Linnik's constant is at most $3.99$

Introduction

The result

For an integer $q \ge2$ and an integer $a$ with $(a,q)=1$, let $P(a,q)$ be the least prime congruent to $a$ modulo $q$. Dirichlet’s theorem [31] ensures that this prime exists. Linnik [77, 78] proved that there are absolute constants $C$ and $L$ such that

for every reduced residue class. An exponent $L$ for which such a constant $C$ exists is called admissible; Linnik’s constant is the infimum of the admissible exponents.

Theorem 1.1 (Least prime in a reduced residue class). There is an absolute constant $q_{0} \ge2$ such that, for every integer $q \ge q_{0}$ and every integer $a$ with $(a,q)=1$, there is a prime $p$ satisfying

Consequently, there is an absolute constant $C>0$ such that $P(a,q) \le Cq^{3.99}$ for all $q \ge2$ and $(a,q)=1$. In particular, Linnik’s constant is at most 3.99.

We do not compute $q_{0}$ or $C$; the issue of effectivity is discussed at the end of Section 13. Except in the presence of an extremely close real zero, the proof finds a prime in the smaller interval

The lower endpoint comes from the support of the weight used to detect primes. When a real zero is extremely close to 1, Heath-Brown’s theorem on Siegel zeros [50] instead gives $P(a,q) \le q^{3.5}$.

The argument combines published analytic estimates, new lemmas proved here, and exact finite computations. The theorem requires all three parts. In particular, checking a numerical certificate proves a bound for its specified configuration of zeros; the analytic argument must also show that every modulus gives a configuration covered by one of the certificates. We make this passage explicit in Sections 11 and 13.

Earlier work

The admissible exponents in Table 1 record the main numerical developments. Heath-Brown [51] obtained 5.5, and Xylouris [143, 144, 145, 146] subsequently obtained 5.2, 5.18, and 5. Before them, Pan [98, 99] gave the first numerical values, and Chen, Jutila, Graham and Wang lowered them in a long series of papers, which includes Graham’s introduction of Selberg sieve weights into zero-density estimates [39, 41]. Our proof uses the framework of these works. It rests on the three principles that Heath-Brown isolates at the start of his paper: a zero-free region with at most one exceptional zero [27, 43, 74, 97], the repulsion of other zeros by an exceptional zero, known as the Deuring–Heilbronn phenomenon [30, 53, 78], and a log-free zero-density estimate [77]. The last means that the density bound has no additional power of $\log q$, a feature needed when studying zeros at distance $O(1/\log q)$ from 1. Simpler proofs of these principles, and of Linnik’s theorem, were given by Rodosskiĭ [111], Turán [133, 135], Knapowski [71], Fogels [32], Gallagher [37], Jutila [65], Motohashi [91, 92] and Bombieri [10], among others; accounts are in the books [110, 84, 62, 34].

$L$	Year	Author(s)	Reference
10000	1957	Pan (announced)	[98]
5448	1958	Pan	[99]
777	1965	Chen	[18]
630	1971	Jutila (reported by Turán)	[134]
550	1970	Jutila	[64]
168	1977	Chen	[19]
80	1977	Jutila	[66]
36	1977	Graham	[39]
20	1981	Graham	[41]
17	1979	Chen	[20]
16	1986	Wang	[139]
13.5	1989	Chen and Liu	[21, 22]
11.5	1991	Chen and Liu	[23]
8	1991	Wang	[140]
5.5	1992	Heath-Brown	[51]
5.2	2009	Xylouris	[143]
5.18	2011	Xylouris	[144]
5	2011	Xylouris	[145, 146]
3.99	2026	this paper

Type	First character and zero	$n$	$\alpha$
rr	$\chi_{1}$ real, $\rho_{1}$ real	$1$	$1$
rc	$\chi_{1}$ real, $\rho_{1}$ nonreal	$1$	$2$
complex	$\chi_{1}$ nonreal	$2$	$1$

The conjectured scale is much smaller. Chowla [24] conjectured $P(a,q)\ll_{\varepsilon}q^{1+\varepsilon}$ for every $\varepsilon>0$; the maximum over reduced classes is conjecturally of order $\varphi(q)(\log q)^2$ [138, 109, 42, 76]. The Generalized Riemann Hypothesis gives admissible exponents $2+\varepsilon$; see [4, 73, 15] for explicit conditional bounds. Explicit [9] and uniform [128] unconditional estimates for primes in progressions are also known. In the other direction, an exceptional (Siegel) zero helps: Heath-Brown [50] showed that a sufficiently strong exceptional zero forces $P(a,q)\leq q^{3+\delta}$, in an effective form that we use below, and $q^{2+\delta}$ ineffectively; see also [33, 61, 147]. Other proofs of Linnik’s theorem use pretentious methods [114], sieve methods [36], or avoid $L$-functions [80], but give larger exponents. For moduli with special multiplicative structure there are stronger zero-free regions [7, 38, 59], and much smaller exponents are known [16].

From zeros to primes

Write $\mathcal{L}=\log q$ and express a zero of a Dirichlet $L$-function as

Thus small $\lambda$ means that the zero is close to the line $\sigma=1$; $\mu$ is its height multiplied by $\log q$. We group each nonprincipal character with its complex conjugate and call this a family. The parameters $\lambda_{1},\lambda_{2},\lambda_{3}$ record the first zero of successive families in a specified rectangle near 1. The parameter $\lambda^{\prime}$ records the next zero occurrence in the first family, after removing the distinguished zero and its conjugate where appropriate. Precise definitions, including multiplicities and ties, are given in Section 2.

A nonnegative weight $h$, supported in $[A,L]$ with $A=L-2T$ and $T=0.416829$, detects primes between $q^{A}$ and $q^{L}$. Let $H$ be its Laplace transform. Xylouris’s positivity criterion [145] (3.57) gives

where $\eta=10^{-6}$ and

Here $R_{P}$ is a fixed square in the normalized coordinates, and zeros are counted with multiplicity. It is therefore enough to prove $W\leq1-2\eta$. The contribution of an individual zero decays exponentially with $\lambda$, so zeros close to 1 require the most accurate estimates.

We combine three kinds of information about these zeros. First, a per-character bound estimates the total contribution of one $L$-function from a lower bound for the parameters of its zeros. Second, a far-density bound limits a weighted sum over characters, with weight depending on a selected zero of each character. Third, near-density bounds constrain the characters whose selected zeros lie closest to 1. Zero-location estimates for $\lambda_{1}$, $\lambda'$, $\lambda_{2}$, and $\lambda_{3}$ determine which of these bounds are available in a given case.

The new near-density estimate.

Heath-Brown’s near-density estimate [51] bounds the number $N(\lambda)$ of characters with a zero in $\sigma\ge1-\lambda/\mathcal{L}$, $|t|\le1$. Its proof applies the Cauchy–Schwarz inequality to a prime sum at the single point $\beta_{1}=1-\lambda_{1}/\mathcal{L}$, and compares the resulting Gram form of the characters with the diagonal. Because every character is tested at the same point, a zero at distance $\lambda$ counts only through the single number $N(\lambda)$, and the bound is weaker than the density estimate of Heath-Brown’s §11 as soon as $\lambda\ge1.1$ and void soon after. That density estimate, proved with Selberg sieve weights after Graham, has the form eq:11.4 of a sum over characters of a weight that depends on each character’s own distance $\lambda^{(k)}$, and it contains the count eq:11.5 of the characters with $\lambda^{(k)}\le\lambda$. In his closing list of possible improvements Heath-Brown writes [51]:

It would be nice to have a weighted version of Lemma 12.1, in the way that eq:11.4 is a weighted version of eq:11.5. Unfortunately, no neat way of achieving this seems available.

The graded near-density lemma (Theorem 8.4) supplies such an estimate. We test each character at its own anchor, a real parameter determining the point at which a smoothed prime sum is evaluated. Cauchy–Schwarz relates these prime sums to a Gram form of the characters. We estimate its off-diagonal terms by the local explicit formula and, where sieve weights are used, Burgess’s bound; Graham’s sieve asymptotic controls the diagonal terms. Allowing the anchors to vary is the grading in the name of the lemma.

After normalization, the resulting constraint has the form

Here $x_{+}=\max(x,0)$, the index $j$ runs over character–height entries, $v_{j}$ is a lower bound for the response of an entry, and $D_{j}>0$ and $d>0$ bound its diagonal and common correlation costs. Most entries correspond to one selected zero; some retain a second zero of the same character, with an explicit correction for their correlation.

Proposition 8.8 turns this inequality into a form suited to computation: there is a number $\tau\in[0,\sqrt{d}]$ such that

A single constraint now distinguishes zeros at different distances from 1. The sieve weighting has precedents in Graham’s work [39, 41], Heath-Brown’s far-density argument, and the sieve-weighted large-sieve inequalities of Motohashi [91].

Where the new lemma enters. In Heath-Brown’s proof the near-density estimate enters at the very end, in the assembly of [51], §15. There the characters with a zero below a level $\Lambda$ are sorted into bins $(\Lambda_{r+1},\Lambda_{r}]$, with $\Lambda_{r}=\Lambda-0.025r$; each character is charged the cost of the end of its bin nearer to $s=1$; and the numbers of characters in the bins are controlled only through the counts $N(\Lambda_{r})$, bounded one level at a time by Lemma 12.1 (his Table 13). The far-density lemma controls the characters beyond $\Lambda$. Xylouris’s assembly [145], §6 has the same structure, with a sharper form of Lemma 12.1 (his Lemma 5.3). A count $N(\lambda)$ treats all characters with a zero below $\lambda$ alike, so it cannot tell a few zeros very close to $s=1$ from many zeros further away. In our proof this place is taken by the near rows of the leaf programs described below. Each near row is one instance of Theorem 8.4: a single quadratic constraint in which every bin of characters enters with its own feature, computed at its own anchor. The weighted version of Lemma 12.1 that Heath-Brown asked for is thus used exactly where his count was used, in every leaf program, and it is the main source of the improvement (Section 1.6).

The finite case analysis.

We divide the middle range $0.1\le\lambda_{1}<1.5$ into cases determined by the type of the first family and intervals for the first few zero parameters. There are four levels in the computation, serving different purposes.

1. A root case specifies a range of possible zero configurations. The 58 parent rows taken from published zero-location tables lead to 4453 root cases, covering the middle range.

2. A leaf is a terminal case after further subdivision and valid zero-location exclusions. There are 4788 leaves that require numerical bounds.

3. A leaf program groups characters into bins according to their selected zero parameters. Its nonnegative variables are the numbers of characters in these bins, together with weighted masses for the tails. The objective bounds $W$, and its constraints express the far-density bound, restrictions on the first families, and the near-density inequalities. For fixed thresholds these constraints are linear.

4. A threshold box bounds the auxiliary numbers $\tau$ in the near rows. On each box, a linear relaxation is either excluded or bounded by exact dual multipliers. Subdividing these boxes covers every possible choice of thresholds.

The case tree partitions information about zeros; the threshold boxes partition the auxiliary parameters used to bound them. Keeping these two subdivisions distinct is essential to the coverage argument.

Each leaf, or each subcase of a leaf, has its own certificate, a binary tree of threshold boxes. The 4788 leaves give 5591 such certificate trees, because some leaves are subdivided further, and these trees contain 4,196,879 threshold boxes. On each box the near rows are replaced by linear relaxations, some with two alternative forms, and every choice of alternatives, a relaxation case, is certified separately: the boxes carry 29,397,336 relaxation cases, of which 9430 are shown to be infeasible and the others are bounded by their own integer dual multipliers. The largest certified value is $0.9999998223\ldots< 1$; it bounds the zero sum together with the required error allowance. All certificate inequalities are checked in integer arithmetic. The subdivisions and zero-location arguments of the tree do not depend on the target exponent; only the leaf programs do. Table 2 compares these numbers with the case analyses of Heath-Brown and Xylouris, and Table 10 lists every level of the analysis.

	[51]	[145]	This paper
	$L=5.5$ (1992)	$L=5$ (2011)	$L=3.99$
Near-density input	counts $N(\lambda)$ from Lemma 12.1	counts $N(\lambda)$ from a sharper Lemma 12.1	the weighted (graded) lemma, Theorem 8.4
Numerical tables	12	11 more	58 rows from theirs, plus over 100 new
Final case analysis	14 intervals of $\lambda_1$	21 cases, each split by two zero counts	4453 roots, 4788 leaves
How a case is closed	floating point, without rounding analysis	Maple, 10-digit floating point	exact certificates on 4.2 million boxes
Largest value of $W$ (must be below 1)	0.9943	0.998	0.99999982

Level	Number	What it partitions or represents	Where defined	How it is checked
Parent rows	58	Published implications: type t and $\lambda_{1} \in[A_{j}, B_{j}]$ give $\lambda' \ge p_{j}$, $\lambda_{2} \ge r_{j}$ (33 rr, 13 rc, 12 complex)	Proposition 7.5	Printed tables; Heath-Brown’s Tables 4 and 7 recomputed
Base cells	340	Parent intervals cut into cells of width 0.01 (145, 89, 106)	Section 11.3	Validated in the replay
First-zero cells	478	Tiling of the base cells (195, 115, 168), with bounds $p \le\lambda'$ and $l_{2} \le\lambda_{2}$	Section 11.3	Validated in the replay
Gap cases	684	422 cells without a gap; 262 gaps for $\lambda'$ partitioning 56 complex cells	Section 11.3	Validated in the replay
Specifications	2768	Gap cases with a second-family record: 1083 rr (195 unreserved, 444 + 444 reserved), 115 rc, 1570 complex	Definition 11.2	Validated in the replay
Root cases	4453	2768 specifications inside, and the 1685 not of type rr also outside	Theorem 11.4	Hand proof; replay
Refinement nodes	1959	1029 splits, 236 restricting location updates, 694 exclusions (75 by real location updates, 422 by lower bounds for $\lambda_{2}$ or $\lambda'$, 197 by positivity); $4453 + 1029 = 4788 + 694$	Section 11.4, Table 9	Replay; some location rows by separate programs
Leaves	4788	3146 inside (233 supplementary) and 1642 outside; 343 inside and 70 outside roots have none	Section 11.4	Replay; Lean numeric checker
Certificate trees	5591	One per leaf or subcase of a leaf: 3949 inside (349 for supplementary leaves), 1642 outside	Section 11.6	Replay; independent checker; Lean checker
Threshold boxes	4,196,879	Terminal boxes of the trees: 4,109,455 inside (1,042,363 for supplementary leaves), 87,424 outside	Definition 10.6	As above
Relaxation cases	29,397,336	$2^{k}$ in a box with $k$ tangent rows; 9430 closed by an exclusion, the rest by integer duals	Definition 10.6	As above, in exact arithmetic
Rows and exclusions
Near rows	16,842	All family, shifted and graded rows (13,528 inside, 3314 outside), with 11,135,323 entries	Section 9.5	Interval arithmetic; Lean row checker
Two-test rows	37	On 37 inside roots: 31 single, 2 mixture, 4 paired	Section 9.6	Replay; exact comparison
Complex location rows	94	25 for $\lambda_{2}$ and 69 for $\lambda'$ (Propositions 7.12 and 7.13)	Appendix A	Interval arithmetic; separate program
Degree-5 real rows	22	Cells covering $[0.700, 0.755]$, used in 300 location nodes	Proposition 7.10, Table 6	Interval arithmetic; replay
Positivity exclusions	197	Reserved nodes of type rr; least normalized margin $7.8 \cdot10^{-5}$	Proposition 7.14	Interval arithmetic; replay

Triples of numbers are counts for the types rr, rc, complex; “supplementary” refers to the supplementary leaves of Section 11.4. Each split adds one terminal node; hence the identity in the refinement row. The printed tables were compared with the pages of [51, 145], and Heath-Brown’s Tables 4 and 7 were recomputed in floating point (Section 7.1). The replay does not regenerate the 94 complex rows or the conditions [145] (4.29), (4.34); separate interval programs check them. The Lean row checker covers all near rows except the 37 two-test rows.

Node	Inside	Outside
first-zero / second-family / gap splits	1 / 408 / 593	0 / 27 / 0
location updates, real (restrict / exclude)	225 / 75	–
location updates, complex (restrict / exclude)	11 / 0	–
exclusions by lower bounds for $\lambda_{2}$ or $\lambda'$	352	70
exclusions by positivity	197	0
leaves	3146	1642

The exterior ranges are shorter arguments. An exceptionally small $\lambda_{1}$ is covered by Heath-Brown’s theorem on Siegel zeros. The rest of $\lambda_{1} \le0.1$ is handled by his quantitative Deuring–Heilbronn estimates and zero-location tables. A single further certificate treats $\lambda_{1} \ge1.5$. If the relevant rectangle contains no zero, then $W = 0$ and the positivity criterion applies immediately.

Comparison with Heath-Brown and Xylouris

Our framework is Heath-Brown’s [51] with Xylouris’s refinements [145, 146]. We use the following from their work:

• the positivity criterion [145], (3.57);

• the per-character bound [145], Lemma 3.10;

• the far-density lemma [145], Lemma 5.1;

• the zero-free regions and Deuring–Heilbronn estimates, as printed in their tables;

• the local explicit formulas [145], Lemmas 3.1, 3.2, which are [51], Lemmas 5.3, 5.2.

The case analyses compared. Both earlier proofs are computer-assisted, and both are organized, like ours, as a case analysis on the first zeros; Table 2 compares the three. Heath-Brown’s paper has twelve numerical tables of zero-location and density bounds. Its final assembly [51], §15 treats $\lambda_{1} \ge0.348$ in fourteen intervals, mostly of length 0.02, each by a chain of inequalities evaluated numerically; the largest total it reports is 0.9943. Heath-Brown writes [51], §1 that “the arguments rely heavily on numerical calculations. These we do not reproduce in full, nor have we attempted any rigorous analysis of the rounding and truncation errors in the computer algorithms employed.” Xylouris’s dissertation adds eleven tables. Its final assembly [145], §6.4, pp. 86–87 has 21 cases: three intervals of $\lambda_{1}$ for a real zero of a real character, two for a complex zero of a real character, and sixteen for a nonreal character. Each case is split further according to the possible values of two zero-counting functions [145], (6.46), and the computations were done in Maple with ten digits [145], p. 14; the largest value is $W < 0.998$. In both, each case is closed by a chain of inequalities in which the unknown zeros are estimated separately, one after another.

In our proof the same role is played by 4453 root cases and 4788 leaves, and each leaf is closed by a linear program with an exact certificate instead of a floating-point evaluation. The 4.2 million boxes have no counterpart in the earlier work. They do not subdivide the configurations of zeros. They subdivide the auxiliary thresholds of the near rows, and they arise because the graded lemma is a quadratic constraint rather than a count. The largest certified value, 0.99999982 against 0.9943 and 0.998, shows how much finer the analysis must be at $L = 3.99$.

Where the gain comes from. The gain from 5 to 3.99 has three sources:

• the graded near-density lemma, which replaces the counts $N(\lambda)$ and applies to every range of $\lambda$ at once;

• the linear-programming combination, which lets all constraints act together on each case instead of chaining worst cases;

• a fine case analysis of the first zeros, possible because every case is certified by machine.

The last source alone does not go far. Xylouris estimates that his method, with much larger computations (smaller intervals and finer grids), would give about 4.96 [146], p. 82.

The certified exponent and numerical evidence.

We state the result at 3.99 because this is the exponent for which the complete cover has been certified. The tightest cases have a nonreal first character and $\lambda_1 \in[0.64,0.80]$. In this range the available lower bounds for $\lambda_2$ are comparatively small, which restricts the anchors in the near-density estimates. Several of the corresponding leaf bounds are within $10^{-7}$ of 1 after the error allowance, and require fine subdivisions.

Exploratory floating-point computations suggest that a somewhat smaller exponent may be accessible. One difficult leaf, with the reserved-family columns of Section 11.5, has computed values 0.992 at $L=3.97$ and 1.020 at $L=3.95$. These are values of particular relaxations, not certified global bounds or an obstruction to other methods. Similarly, the positive numerical margins for the exterior ranges at smaller exponents do not constitute a complete proof there. The verification reported in this paper is at $L=3.99$.

Verification and a guide to the proof.

The certificates have been checked by independent programs. Parts of the analytic argument and the soundness of the checkers have also been formalized in Lean, with the published analytic inputs stated as hypotheses. This formalization includes a theorem that a valid certified leaf yields a prime when the modulus satisfies that leaf’s hypotheses about zeros. It does not include the full passage from arbitrary moduli to leaves, or the exterior arguments. Section 14 gives the verification records and separates these claims precisely.

Sections 2 and 3 give the notation and published inputs. Sections 4 to 6 reduce prime detection to bounds for the zero sum, and Section 7 supplies the zero-location estimates. The main new analytic argument is in Section 8; Section 9 derives the constraints used in the computation.

Section 10 proves certificate soundness, while Section 11 shows that the zero configurations are covered and satisfy the programs. The exterior cases and the final choice of constants are treated in Sections 12 and 13. A reader interested first in the new density estimate may read Sections 2 and 3 and then Section 8, returning to the numerical construction afterwards.

Figure 1 shows how the main results depend on one another, and which of them are published inputs, new results, computations or formally verified statements. The terms of the case analysis are introduced in Section 2.4, and the fixed constants and the error budget in Section 2.7. Table 10 in Section 11 lists all levels of the finite analysis, and Table 12 in Section 14 records, for each component, its source and how it was checked.

Component	Source	Kind	Method	Checked by
Positivity criterion and detection (Input 4.2 and Proposition 4.4)	Published [145]	Analytic	–	Lean, with the criterion as a hypothesis
Envelope and first-family bounds (Section 5)	New, after [145]	Analytic	Interval arithmetic	Hand proof; a hypothesis in Lean, which checks only the enclosures
Far density (Input 6.1)	Published [145]	Analytic	–	Lean hypothesis; profile conditions proved in Lean
Far constants $V$, $w$ (Section 6)	New profiles	Numerical	Interval arithmetic	Multiple-precision audit; Lean numeric checker
Printed zero-location tables (Proposition 7.5)	Published [51, 145]	Analytic	As printed	Compared with the pages; Heath-Brown’s Table 4 and 7 recomputed in floating point
New location rows (Section 7)	New	Both	Interval arithmetic	Replay; the 94 complex rows by a separate program
Conditions [145], (4.29), (4.34)	Published; new check	Numerical	Interval arithmetic	Separate programs
Conductor refinement $\frac{1}{108}$ (Section 7.6)	New	Analytic	–	Hand proof
Graded near lemma, response lemma, threshold form (Section 8)	New	Analytic	–	Lean (Comparator, two kernels)
Family, shifted and graded rows (Section 9)	New	Numerical	Interval arithmetic	Lean row checker (16,842 rows); multiple-precision audit
Two-test rows (Section 9.6)	New, after [145]	Both	Interval arithmetic	Hand proof; replay; exact comparison with a separate implementation on every certificate interval (constants: that implementation’s interval code only)
Certificate soundness (Theorem 10.7)	New	Analytic	–	Lean (Comparator, two kernels)
Leaf certificates (Definition 10.6)	New	Numerical	Exact integer	Replay; independent checker; Lean checker
Leaf data (Section 11.5)	New	Numerical	Interval arithmetic	Multiple-precision audits; Lean numeric checker (count-type integers: audit only)
Case tree and realization (Section 11)	New	Analytic	–	Hand proof; replay
Tiny exceptional zero (Proposition 12.1); empty rectangle	Published [50]; $W=0$	Analytic	–	As printed
Small exceptional zero (Theorem 12.5)	New, from lemmas of [51]	Both	Interval arithmetic	Hand proof; exterior replay; margins recomputed independently
Large first zero (Theorem 12.6)	New	Numerical	Exact integer	Replay’s exact checker; independent checker; Lean certificate (native and kernel), numeric and row checkers
Assembly (Section 13)	New	Analytic	–	Hand proof

Notation and conventions

We use standard notation for Dirichlet characters and their $L$-functions. For background on Dirichlet $L$-functions and their zeros we refer to [57, 131, 69, 85, 26, 62, 87, 124], and for the history of the subject to [94]. The distinctions between a zero and a zero occurrence, and between normalized and physical height, will be used throughout the proof.

Moduli, characters and zeros

Throughout, $q \ge2$ is an integer modulus and

Characters are Dirichlet characters modulo $q$; $\chi_{0}$ is the principal character. For a nonprincipal character $\chi$ we consider the zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$ with $\beta>0$, counted with multiplicity, and we write

We call $\lambda=\lambda_{\rho}$ the parameter of $\rho$ and $\mu=\mu_{\rho}$ its normalized height; $\gamma=\gamma_{\rho}$ is its physical height. Thus a height bound always concerns $|\gamma|$ or $|\mu|$. Smaller $\lambda$ means a zero farther to the right. The multiplicity of $\rho$ as a zero of the specified $L(s,\chi)$ is $m_{\rho}$. A zero occurrence is one copy of the pair $(\chi,\rho)$: a zero of multiplicity $m_{\rho}$ gives $m_{\rho}$ occurrences. Unless stated otherwise, sums over zeros count multiplicity. A zero of an imprimitive character is a zero of the $L$-function of the primitive character inducing it, with the same multiplicity, as long as $\beta>0$; all zeros considered below have $\beta>1/2$.

Regions

Following [145] (3.7) we put, for $x>0$,

There is an integer $l=l(q)$ with $1\leq l\leq\mathcal{L}/10$ such that no zero of any $L(s,\chi)$ lies in $R(10l)\setminus R(l)$ (Input 3.5). We fix one such choice, for example the least admissible integer. The distinguished zeros (Definition 2.2) and the representatives (Section 2.4) are selected in $R(l)$. The local explicit formula also involves disc zeros, which need not all belong to this rectangle. Three further regions have separate roles:

• the prime window $R_{P}=\{0\leq\lambda\leq C_{P},\ |\mu|\leq C_{P}\}$, whose zeros are the only ones that enter the zero sum;

• the buffer $R_{B}=\{0\leq\lambda\leq C_{B},\ |\mu|\leq C_{B}\}$, with $C_{B}>C_{P}+M$;

• the height-one region $|\gamma|\leq1$, in which the far-density lemma operates.

The constants $C_{P}$, $M$ and $C_{B}$ are fixed independently of $q$, in the order described in Section 13. For sufficiently large $q$, $R_{P}\subset R_{B}\subset R(l)$, and the buffer has physical height $C_{B}/\log q<1$. The word inside means that the selected first zero belongs to $R_{B}$; outside means that it does not. It does not refer to $R(l)$.

Families and the first zeros.

Definition 2.1 (Families and successive minima). A family is the set $\mathcal{F}(\chi)=\{\chi,\overline{\chi}\}$ for a nonprincipal character $\chi$. It has one member when $\chi$ is real and two otherwise. If there is a nonprincipal zero in $R(l)$, select a pair $(\chi_{1},\rho_{1})$ with least parameter and write $\lambda_{1}=\lambda_{\rho_{1}}$, $\mu_{1}=\mu_{\rho_{1}}$, and $\mathcal{F}_{1}=\mathcal{F}(\chi_{1})$. Remove all occurrences belonging to $\mathcal{F}_{1}$ and repeat to define $(\chi_{2},\rho_{2})$, $\lambda_{2}$, and $\mathcal{F}_{2}$; continue in this way. Thus $\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\cdots$. We call $\mathcal{F}_{1}$ the first family, $\mathcal{F}_{2}$ the second family and $\mathcal{F}_{3}$ the third family, and $(\chi_{1},\rho_{1})$ the first zero. If a subsequent minimum is over an empty set, its value is $+\infty$. Ties may be resolved by any admissible choice; each result used below holds for all such choices. If the initial set is empty, we use the empty-region argument of Section 12.

These are the selections of Xylouris [145]. They order families, not the individual zeros of one $L$-function.

Definition 2.2 (The additional first-family zero). The distinguished occurrences are one copy of $\rho_{1}$ as a zero of $\chi_{1}$, together with one copy of $\overline{\rho_{1}}$ as a zero of $\chi_{1}$ if $\chi_{1}$ is real and $\rho_{1}$ is not, or as a zero of $\overline{\chi_{1}}$ if $\chi_{1}$ is not real. We write $\lambda'$ for the least parameter of a zero occurrence of $\chi_{1}$ or $\overline{\chi_{1}}$ in $R(l)$ that is not distinguished; thus $\lambda'=\lambda_{1}$ when $\rho_{1}$ is a multiple zero. If no occurrence remains, put $\lambda'=+\infty$.

When $\lambda'$ is finite, it can always be realized by a zero of $\chi_{1}$ itself. Indeed, zeros of $L(s,\overline{\chi_{1}})$ are conjugates of zeros of $L(s,\chi_{1})$, with the same multiplicities. Replace any minimizing occurrence of $\overline{\chi_{1}}$ by its conjugate occurrence of $\chi_{1}$. We call every resulting non-distinguished occurrence $\rho'$ with parameter $\lambda'$ admissible (a second copy of $\rho_{1}$ is admissible when $\rho_{1}$ is a multiple zero), and write $\gamma'$ for its height; as $\rho'\in R(l)$, $|\gamma'|\leq l$. The arguments below hold for every admissible choice; in the second-zero split of Section 11.6 the choice is the one used there.

The following table fixes the three types and the two counts used in the first-family bounds. Here $n=|\mathcal{F}_{1}|$, while $\alpha$ counts distinguished occurrences per character.

For type rc we choose the conjugate with $\mu_{1}>0$.

The vocabulary of the case analysis.

In the middle range $0.1\leq\lambda_{1}<1.5$ the proof is a finite case analysis, defined precisely in Sections 10 and 11. Its terms are needed in the analytic sections that precede those, and we introduce them here.

• Anchor. The local explicit formulas of Section 3 are applied at points $s=1-\sigma/\mathcal{L}+i\gamma$; the real number $\sigma$ is the anchor. An inequality anchored at $\sigma$ is useful only if the zeros near $s$ with parameter below $\sigma$ are absent or are retained explicitly, so anchors are chosen at or below known lower bounds for zero parameters.

• Specification. A specification $\mathfrak{s}$ (Definition 11.2) records the type of the first family; an interval $[a,b]$ containing $\lambda_{1}$, the first-zero cell; a lower bound $p\leq\lambda'$ and possibly an interval $[\mathrm{lo}',\mathrm{hi}']$ containing $\lambda'$, called a gap; a lower bound $l_{2}\leq\lambda_{2}$; a lower bound $r$ for the parameters of the zeros of height at most $1$ of the families other than the first; possibly a reservation (next item); and whether $\rho_{1}$ lies inside or outside the buffer. Its configuration set $C(\mathfrak{s})$ consists of the zero configurations compatible with these data.

• Reserved and ordinary characters. A reservation concerns a family other than the first whose least parameter among zeros of height at most $1$ is as small as possible. It fixes the

number $n_{2}\in\{1,2\}$ of its characters and an interval $[\operatorname{lo}_{2},\operatorname{hi}_{2}]$ for that parameter. This reserved family need not be $F_{2}$. The characters outside the first and the reserved family are ordinary.

• Representatives. The height-one representative of a nonprincipal character $\chi$ is a zero of $L(s,\chi)$ in $R(l)$ with $|\gamma|\leq1$ and least parameter; its $T^{*}$-representative is defined in the same way with $|\gamma|\leq T^{*}$, for a height $T^{*}\in[\frac{1}{2},1]$ fixed in Lemma 6.3 (Definition 6.4). The leaf programs place a character by the parameter of one of its representatives.

• The case tree. The roots are 4453 specifications whose configuration sets cover the middle range. Each root is refined by splitting intervals and by applying zero-location results. This gives a finite tree of specifications, the root’s refinement tree; its vertices are nodes, and together these trees form the case tree. A node at which a zero-location result sharpens one of the bounds is a location update; a node whose configuration set is shown to be empty is excluded. The terminal nodes that are not excluded are the leaves. Some leaves are divided further into finitely many subcases.

• Leaf programs, bins and columns. Each leaf or subcase has a leaf program (Definition 10.1) whose value bounds $W$ for every configuration in it. The parameters of the ordinary characters (and in some leaves those of the reserved characters) are divided into finitely many intervals, the bins. The column of a bin is the variable of the program that counts the characters whose representative zero (Definition 6.4) has parameter in the bin. The ordinary characters beyond the last bin form the tail, whose column is a weighted mass rather than a count. In an outside leaf the first family is represented by hidden columns, which bin the least parameter $t_{\chi}$ of the zeros in $R_{P}$ of each first-family character (Section 5.7). The constant first bounds the cost of the first family in an inside leaf (0 in an outside leaf), plus any fixed charge for the reserved family, and final is an allowance for the error terms.

• Far budget. The far-density lemma (Input 6.1) bounds a sum over all characters of a positive decreasing function $w$ of the parameter of a selected zero of height at most 1. We call $w(\lambda)$ the far weight of such a zero, and the bound (9) the far budget; in a leaf program it is one linear constraint.

• Near rows. The other constraints of a leaf program are conditions on counts and the near rows. A near row is, with one exception (Section 9.6), an instance of the graded lemma (Theorem 8.4) in the form of Proposition 8.8. Its terms are entries, one for each character (occasionally each zero) that it includes. Each entry has a feature, a lower bound for how strongly its zero is detected, and a diagonal, what it costs on its own, and one correlation term bounds the interaction between distinct entries (Proposition 8.8). The entries of a family whose zeros are known are grouped into family terms. A row is built from two test functions, a detector and a Gram test (Definition 8.1); by the choice of its anchors and sieve weight it is a family, shifted, graded or two-test row (Table 8). It is linear in the columns once an auxiliary number, its threshold $\tau$, is fixed.

• Certificates. A certificate of a leaf or subcase is a binary tree, the certificate tree, whose nodes are boxes of thresholds, starting from a box that contains every admissible threshold; its leaves are the terminal boxes. On a terminal box each near row is replaced by a linear relaxation, and each choice among the alternatives of these relaxations, a relaxation case, is either shown to be infeasible (an exclusion) or bounded below 1 by integer dual multipliers, in exact integer arithmetic (Section 10).

Row	Reference anchor $s$	Sieve weights	Entries, inside	Entries, outside
family	$s_{1}$	none $(\Xi=0)$	(a), (f) and (b), (c) or (d)	(a), (f), (g)
shifted	$\min(1.9,p^{*},l_{2})$	none $(\Xi=0)$	(a), (e), (f)	not used
graded	1.5, 1.6, 1.9, 2.1 or 2.3	present	(a), (b), (f)	(a), (f)
two-test	the shift of the leaf	two test functions	Theorem 9.4, on 37 roots	not used

Elementary notation

For real $x$ we write $x_{+}=\max(x,0)$, and $\mathbf{1}_{E}$ denotes the indicator of a condition or set $E$. The symbol $\varphi(q)$ is Euler’s totient; the separately defined $\varphi(\chi)$ is a conductor coefficient (Section 3). A cost means an upper bound for a contribution to the normalized zero sum $W$; a budget is the right-hand side of a constraint on these contributions. All finite decimals used as parameters are exact rational numbers unless an approximation is explicitly indicated.

Asymptotic conventions

Statements about zeros below are understood for all sufficiently large $q$, unless a different range is stated. The threshold $q_{0}$ depends on finitely many fixed choices (the exponent, test functions, weights, constants and certificates, and the tolerances of the published results we use); the order in which these are chosen is spelled out in Section 13. We write $o(1)$ for a quantity tending to $0$ as $q \to\infty$, uniformly in everything except the fixed choices. The number

is a fixed tolerance used throughout, and the exponent is

Certificate coefficients use the scale $S = 10^{16}$. Counts of characters are unscaled; the precise conversion between coefficients and real quantities is given in Section 10.1.

Constants and the error budget

The fixed constants of the proof, with their values, their roles and the stage of Section 13.1 at which each is chosen, are listed in Table 15 in Appendix B. The tolerance $\eta= 10^{-6}$ is spent as follows; the allowance $5\eta$ in each leaf value covers the analytic errors with room to spare.

Symbol	Value	Role	Where fixed	Stage
Exponent and kernel
$L$	$3.99$	The exponent	Section 2	(S1)
$T$	$0.416829$	Support of $\psi$ is $[0,T]$; $h$ is supported in $[L-2T,L]$	Section 4.1	(S1)
$\kappa$	$T/16=0.0260518125$	Width of the 16 steps of $\psi$	Section 4.1	(S1)
$b_0,\ldots,b_{15}$	Table 3	Step heights of $\psi$	Table 3	(S1)
$A$	$L-2T=3.156342$	Lower end of the support of $h$; primes are found in $(q^A,q^L)$; $A>3$ and $A>2x$	Section 4.1, Lemma 6.2	–
$\Psi(0)$	$\kappa\sum_i b_i=0.1936422928\ldots$	Integral of $\psi$	Section 4.1	–
$H_0$	$\Psi(0)^2=0.0374973375\ldots$	$H(0)$; normalizes the zero sum $W$	(4.1)	–
Tolerances
$\eta$	$10^{-6}$	Detection needs $W\leq1-2\eta$; far budget $(1+\eta)V$; allowance $5\eta$	Section 2, Proposition 4.4	(S1)
$\eta'$	$10^{-6}$	Tolerance of the near rows: in every feature, diagonal and correlation term	Section 9.2	(S1)
Symbol	Value	Role	Where fixed	Stage
$\varepsilon_{1}$	$\eta/23$	Envelope error $\varepsilon_{1}e^{-A\lambda}$ per character; $23=19+4$, where $19>K_{\mathrm{far}}$ bounds $\sum_{\chi}e^{-A\lambda_{\chi}}$ over the ordinary characters and 4 counts the first and reserved characters	Section 5.8	(S1)
$\mathrm{final}/S$	$5\eta$	Allowance included in every leaf value	Section 11.5	(S1)
$\varepsilon$ in Input 4.2	$\eta H_{0}$	Tolerance of the positivity criterion	Proposition 4.4	(S5)
$\varepsilon$ in Input 6.1	$\eta J^{2}$	Tolerance of the far-density lemma; gives (9)	Section 6	(S5)
Analytic constants
$\lambda_{11}$	$0.05$	Least anchor of the envelope; the anchor set $\mathcal{A}$ lies in $[\lambda_{11},C_{P}]$	Section 5.3	(S1)
$s_{\max}$	$3$	Largest anchor of an ordinary entry; $K_{0}$ counts zeros with $\lambda\leq s_{\max}$	Section 6.3	(S1)
$C_{P}$	$\geq\max(C_{0}(\eta H_{0}),3)$	Size of the prime window $R_{P}$; $C_{0}$ is the constant of Input 4.2	(5)	(S2)
$N$	$N(C_{P})$	Bound for the zeros of one character in $R_{P}$	Lemma 5.3	(S3)
$h_{\mathcal{A}},\mathcal{A}$	Mesh with $\lvert B_{\phi}(x)-B_{\phi}(y)\rvert\leq\varepsilon_{1}/4$ for $\lvert x-y\rvert\leq h_{\mathcal{A}}$	Finite anchor set: the grid $\lambda_{11}+h_{\mathcal{A}}\mathbb{Z}$ and the anchors $a,p^{*},p^{\circ}$ of the leaves	Section 5.3	(S3)
$\theta$ (smoothing)	In $(0,1)$, with (5.1)	$L^{1}$-distance of the smooth step function $\psi^{\theta}$ from $\psi$	Section 5.2	(S3)
$c_{\mathrm{out}}$	$C_{P}f_{p}^{\theta}(0)+\lvert(f_{p}^{\theta})'(0)\rvert+e^{C_{P}T}\lVert(f_{p}^{\theta})''\rVert_{1}$, maximized over the anchors $p$	Cost of the distinguished terms of an outside first family	Proposition 5.9	(S3)
$K_{0}$	Independent of $q$	Bound for the zeros with $\lambda\leq s_{\max}$ and $\lvert\gamma\rvert\leq2$	Section 6.3	(S4)
$M$	Large	Height separation for Lemmas 6.3 and 8.7; depends on $K_{0}$ and $\eta_{\min}''$	Section 13.1	(S4)
$C_{B}$	$>\max(C_{P}+M,1.5)$, with $4c_{\mathrm{out}}/(H_{0}C_{B}^{2})\leq\eta$	Size of the buffer $R_{B}$, which separates inside from outside	Section 2, Section 13.1	(S4)
$\delta_{0}$	In $(0,\frac{1}{3}]$	Common disc radius of the explicit formulas	Section 5.3	(S5)
$q_{0}$	Not computed	Maximum of the finitely many thresholds	Section 13.1	(S6)
Far density (profile I; profile II)
$J,\varepsilon_{0}$	$10,\,10^{-7}$	Number of weights $\alpha_{i}$; offset in $w_{0}$ (both profiles)	Section 6	(S1)
$c_{1}$	$0.09035;\ 0.0821922$	Parameter of Input 6.1	Section 6	(S1)
$c_{2}$	$0.235968;\ 0.2170903$	Parameter of Input 6.1	Section 6	(S1)
$\theta_{\mathrm{far}}$	$1.28683;\ 1.4964274$	Exponent in $w_{0}(t)^{2}$	Section 6	(S1)
$\alpha_{1},\ldots,\alpha_{10}$	Table 4	Weights of Input 6.1, of sum 1	Table 4	(S1)
$x$	$1.1736846\ldots;\ 1.1303335\ldots$	Upper end $\frac{2}{3}+3c_{1}+c_{2}$ of the profile; $2x<A$	Lemma 6.2	–
Symbol	Value	Role	Where fixed	Stage
$V$	$175.26640331\ldots$; $243.32098105\ldots$	Right side of Input 6.1 with $\varepsilon=0$; far budget $(1+\eta)V$	Section 6	–
$w(0)$	$10.7054\ldots$; $13.1730\ldots$	Far weight at $\lambda=0$	Section 6	–
$K_{\mathrm{far}}$	$(1+\eta)V/w(0)$: $<16.38$; $<18.48$	Bound for $\sum_{\chi}e^{-A\lambda_{\chi}}$	(10)	–
Leaf programs and certificates
$S$	$10^{16}$	Scale of the stored coefficients	Section 2, Section 10.1	(S1)
$T_{S}$	$10^{6}$	Thresholds are measured in ticks of $1/T_{S}$	Section 10	(S1)
$D_{S}$	$10^{12}$	Scale of the dual multipliers	Section 10	(S1)
$\mathrm{den}$	One of 200, 400, 500, 800, 2000 per leaf	Ordinary bins between $r$ and $R=\max(3,r)$ on the grid $1/\mathrm{den}$	Section 11.5	(S1)
Reserved grid	$1/400$	Grid of the columns of a reserved second family	Section 11.5	(S1)
Near rows
$\varepsilon'$	$1/2000$	Sieve levels $\vartheta_{k}=\frac{1}{2}(t_{k}-\frac{1}{3})=\varepsilon'$	Section 9.2	(S1)
Cells	2000 sieve cells; 4000 cells for $I_{u}$	Equal cells $t_{0}<\cdots<t_{2000}=2\gamma_{f}$; upper Riemann sum for $I_{B}$	Section 9.2	(S1)
$h_{k},\mu_{\mathrm{row}}$	Rational, denominators at most $10^{12}$; $\mu_{\mathrm{row}}=10^{9}$ in family and shifted rows	Sieve heights; they vanish in family and shifted rows	Section 9.2	(S1)
$\lfloor\delta\rfloor_{1/200}$	Grid $1/200$	Offsets at which the diagonal $D(\delta)$ is evaluated	Section 9.2	(S1)
$s_{1}$	$\lfloor50a\rfloor/50$	Safe anchor of a leaf with first-zero cell $[a,b]$	Section 9.3	(S1)
$s$ (shift)	$\min(1.9,p^{*},l_{2})$	Anchor of the shifted row and of the two-test row	Section 9.5, Section 11.5	(S1)
Graded anchors	$1.5$, $1.6$, $1.9$, $2.1$, $2.3$	Reference anchors $s$ of the graded rows used	Section 9.5	(S1)
Exterior regimes
$u_{\mathrm{exc}}$	$\min(1/\eta_{\mathrm{HB}}(\frac{1}{2}),0.08)$	Below it, Proposition 12.1 $(P(a,q)\le q^{3.5})$; not computed	Section 12	(S1)
$K,A_{s}$	$0.1821$; $L-2K=3.6258$	Single triangle $h_{L,K}$ of the small branch	Theorem 12.5	(S1)
$c_{1},c_{2},\phi$	$0.057$, $0.1554$, $\frac{1}{3}$	Parameters of Input 12.4	Theorem 12.5	(S1)
$a_{d},b_{d},V_{d}$	$\frac{3724}{1875}$, $\frac{671}{750}$, $\frac{184750}{6993}$	Exponents and constant of Input 12.4	Theorem 12.5	–
$\alpha$	$1.09<\frac{12}{11}$	Exponent in the bounds $\lambda',\lambda_{2}\ge\alpha\log(1/\lambda_{1})$ on $[u_{\mathrm{exc}},0.08]$	Theorem 12.5	(S1)
$m^{*}$	$\alpha\log12.5=2.7530442\ldots$	Anchor for the other characters on $[u_{\mathrm{exc}},0.08]$	Theorem 12.5	–
$m_{2},m'_{2}$	$2.829$, $4.959$	Bounds $\lambda_{2}\ge2.83-\varepsilon$, $\lambda'\ge4.96-\varepsilon$ with $\varepsilon=0.001$, on $[0.08,0.1]$	Theorem 12.5	(S1)
$B_{1}$	$K^{2}+\frac{K}{4}$	Bound for $B_{1}/4$ for the first character	Theorem 12.5	–
Large branch	$s=s_{1}=\frac{3}{2}$, $\gamma_{f}=1$, $\gamma_{g}=\frac{3}{2}$; $t_{0}=1$	One near row; bins of width $\frac{1}{100}$ on $[1.5,3)$ and a tail	Theorem 12.6	(S1)

1. Detection. With $\varepsilon= \eta H_{0}$ in Input 4.2, the weighted prime sum is at least $\frac{\mathcal{L}H_{0}}{\varphi(q)}(1-W-\eta)$; so $W \le1-2\eta$ suffices (Proposition 4.4).

2. Analytic error. In (8) the error $E$ has two parts (Section 5.8). The envelope errors, which include the smoothing error by (7), total less than $19\varepsilon_{1}$ for the ordinary characters, since $\sum_{\chi}e^{-A\lambda_{\chi}} \le K_{\mathrm{far}} < 19$ by (10), and at most $4\varepsilon_{1}$ for the at most two first-family and two reserved characters; together less than $23\varepsilon_{1}=\eta$. The outside first-family bound (Proposition 5.9) adds at most $4c_{\mathrm{out}}/(H_{0}C_{B}) \le\eta$, by the choice of $C_{B}$ in (S4). Hence $E \le2\eta$.

3. Allowance. Every leaf value includes $\mathrm{final}/S=5\eta$ (Section 11.5); so does the program of Theorem 12.6.

4. Conclusion. By the proof of Proposition 11.6, $W \le\mathcal{V}(x)-\mathrm{final}/S+E \le\mathcal{V}(x)-3\eta$, that is, $W+3\eta\le\mathcal{V}(x)<1$ by Theorem 10.7. Hence $W<1-3\eta$, and Proposition 4.4 applies with a margin $\eta$.

5. Other tolerances. They enter the constraints rather than $E$: $\varepsilon=\eta J_{2}^{2}$ in Input 6.1 gives the far budget $(1+\eta)V$ of (9), the tolerance $\eta'$ is built into every feature, diagonal and correlation term of a near row (Corollary 8.10), and every stored number is rounded in the safe direction (Section 11.5).

Published inputs

This section records the analytic estimates used in the proof. Statements labelled Input are taken from the literature, with notation adapted to this paper. Our main sources are Heath-Brown [51] and Xylouris’s dissertation [145]; the dissertation, rather than Xylouris’s shorter article, is the source of the lemma and equation numbers cited below. Any additional hypothesis needed from a source proof, or extension of a printed statement, is identified explicitly.

Test functions

Definition 3.1 (Conditions 1 and 2). Let $x_{0}>0$ and let $f:[0,\infty)\to\mathbb{R}$ be continuous with $f(t)=0$ for $t\ge x_{0}$.

• $f$ satisfies Condition 1 if it is twice continuously differentiable on $(0,x_{0})$ with $|f''|\le B$ there for some constant $B$ [51, 145].

• $f$ satisfies Condition 2 if $f\ge0$ and its Laplace transform $F(z)=\int_{0}^{\infty}f(t)e^{-zt}\,\mathrm{d}t$ satisfies $\operatorname{Re}F(z)\ge0$ for $\operatorname{Re}z\ge0$ [51, 145].

Condition 1 supplies the regularity needed for the explicit formula. Condition 2 ensures that zero terms with nonnegative real argument can be discarded in an upper bound. The two conditions have different roles, so we specify each one when it is needed.

A function satisfying Condition 1 is bounded, its derivative is bounded on $(0,x_{0})$, and its transform $F$ is entire. The conductor coefficient of a nonprincipal character $\chi$ modulo $q$ is [145]

In particular $\varphi(\chi)=\frac{1}{4}$ for every real character when $q\ge8$ (its order is $2\le\mathcal{L}$), and $\varphi(\chi)\le\frac{1}{3}$ always.

Input 3.2 (Comparison lemma [51, 145]). Let $F_{1},F_{2}$ be holomorphic in $\{\operatorname{Re}z\ge0\}$, with $\operatorname{Re}F_{1}(z)\ge|F_{2}(z)|$ on $\operatorname{Re}z=0$, and suppose that $F_{1}$ and $F_{2}$ tend to 0 uniformly as $|z|\to\infty$ in $\operatorname{Re}z\ge0$. Then $\operatorname{Re}F_{1}(z)\ge|F_{2}(z)|$ for all $\operatorname{Re}z\ge0$.

This is a consequence of the maximum principle [101, 130]. All transforms below are Laplace transforms of bounded functions of compact support [142].

Explicit formulas

The following two lemmas are Heath-Brown’s Lemmas 5.3 and 5.2, as stated by Xylouris [145]. In them $s=\sigma+it$ with

Input 3.3 (Principal character). Let $f$ satisfy Condition 1 and let $s$ satisfy (3). Then

with an implied constant depending only on $f$.

Input 3.4 (Local explicit formula). Let $\chi\ne\chi_{0}$, let $s=\sigma+it$ satisfy (3), and let $f$ satisfy Condition 1 with $f(0)\ge0$. For every $\varepsilon>0$ there are $\delta=\delta(f,\varepsilon)\in(0,1)$ and $q_{0}=q_{0}(f,\varepsilon)$, independent of $\chi$ and $s$, such that for $q\ge q_{0}$

where the sum runs over the nontrivial zeros of $L(s,\chi)$ in the disc, with multiplicity.

These are smoothed forms of the explicit formula [141, 26, 87, 62]; the conductor coefficient $\varphi(\chi)$ comes from Burgess’s bounds, through Heath-Brown’s growth estimate for $L(s,\chi)$ and a Jensen-type formula [51]. Heath-Brown’s printed Lemma 5.2 omits the factor $\Lambda(n)$ on the left, a misprint that Xylouris’s statement corrects. Xylouris notes [145] that the radius $\delta$ and the threshold $q_{0}$ of Input 3.4, and the implied constant of Input 3.3, depend only on $\varepsilon$ and on upper bounds for $x_0$, $x_0^{-1}$, $\sup|f|$ and $\sup_{(0,x_0)}(|f'|+|f''|)$; so both inputs hold uniformly for a family of test functions with such common bounds. In the proof of Heath-Brown’s Lemma 3.1, on which Lemma 5.2 rests, the disc radius is $\delta=\min(1/(2k),\varepsilon_0/(3c_0k^2))$ with $k\geq3$, so $\delta\leq1/6$. The same proof works for every smaller radius, and Heath-Brown remarks after Lemma 5.2 that $\delta=1/\log\mathcal{L}$ may be taken. We may and do assume $\delta<1/2$.

Input 3.5 (The height $l$ [51, 145]). There are $q_0$ and a number $l=l(q)\in\mathbb{N}$ with $l\leq\mathcal{L}/10$ such that for $q\geq q_0$ the function $\prod_\chi L(s,\chi)$ has no zeros in $R(10l)\setminus R(l)$.

The principal $L$-function has no zeros in $R(l)$ for large $q$ [145]. All results we quote from the tables of [51, 145] concern zeros in $R(l)$ for this $l$; Heath-Brown’s proof of Lemma 6.1 produces $l$ and the later sections use it only through $1\leq l\leq\mathcal{L}/10$ and the zero-free annulus.

Input 3.6 (All-zero envelope [145]). Let $C_0$ be the size of the square in Input 4.2, and fix $\varepsilon,\lambda_{11}>0$ and an integer $m\geq1$. For $\lambda\in(\lambda_{11},C_0]$, suppose that:

(i) each $f_{1i}^{\lambda}$, $1\leq i\leq m$, satisfies Condition 1 with support parameter $x_{0i}>0$ and $f_{1i}^{\lambda}(0)\geq0$;

(ii) the quantities $x_{0i}$, $x_{0i}^{-1}$, $\sup|f_{1i}^{\lambda}|$, and $\sup_{(0,x_{0i})}(|(f_{1i}^{\lambda})'|+|(f_{1i}^{\lambda})''|)$ have common bounds depending only on $\lambda_{11}$ and $C_0$, and $f_1^\lambda=\sum_i f_{1i}^\lambda\geq0$;

(iii) $H_2$ is holomorphic in $\operatorname{Re}z\geq0$, $|H_2(\lambda+it)|\leq\operatorname{Re}F_1^\lambda(it)$ for every real $t$, and both transforms tend uniformly to $0$ as $|z|\to\infty$ in that half-plane;

(iv) $\chi\neq\chi_0$ and $L(s,\chi)$ has no zero with $1-\lambda/\mathcal{L}<\beta\leq1$ and $|\gamma|\leq1$.

Here $F_1^\lambda$ is the Laplace transform of $f_1^\lambda$. Then there is an effectively computable $q_0$, depending on $\varepsilon,m,\lambda_{11},C_0$ but not on $\lambda$, such that for $q\geq q_0$

the sum running over the zeros of $L(s,\chi)$ in the square of Input 4.2.

We have included the hypothesis $f_{1i}^{\lambda}(0)\geq0$ in (i), although it is omitted from the printed statement. The proof [145] applies Input 3.2, then Input 3.4 at $s=1-\lambda/\mathcal{L}$ to each $f_{1i}^{\lambda}$, then Input 3.3. It therefore uses $f_{1i}^{\lambda}(0)\geq0$ for each $i$; every application below has this property. The zero-free hypothesis is used only to know that the zeros of $L(s,\chi)$ in the disc of Input 3.4 about $1$ have $\beta\leq1-\lambda/\mathcal{L}$. We use the lemma in this localized form (Section 5).

Character sums and the Selberg sieve.

Input 3.7 (Burgess [51], $k=3$). Let $q\geq1$ and let $\chi$ be a primitive character modulo $q$. Let $N\geq1$ and $1\leq H\leq q$. Then for every $\varepsilon>0$

This is the case $k=3$ of Heath-Brown’s statement, which is Burgess’s theorem [12, 11, 13, 14]; see [52] for an account. It improves on the Pólya–Vinogradov inequality [108, 136] for short intervals. For special moduli stronger bounds are known [7, 16, 6, 70, 100], but we do not use them.

The weights $\xi_d$ in the next input are Selberg’s sieve weights [116, 117] in the form used for zero-density estimates by Graham and Heath-Brown; see [45, 93, 34] for the Selberg sieve in general.

Input 3.8 (Graham [40], p. 84; [51], (11.13), $U=1$). Let $z \ge2$ and $\xi_d=\mu(d)\log(z/d)/\log z$ for $d \le z$, $\xi_d=0$ otherwise. Then for $N \ge z$

Log-free zero density.

Input 3.9 (Log-free density [66], Theorem 1, as quoted in the proof of [51], Lemma 6.1). For $T \ge1$, $\frac{5}{4} \le\sigma\le1$ and every $\varepsilon>0$,

where $N(\sigma,T,\chi)$ counts the nontrivial zeros $\beta+i\gamma$ of $L(s,\chi)$, with multiplicity, in $\beta\ge\sigma$, $|\gamma| \le T$.

Estimates of this kind go back to Linnik [77, 78]; see [133, 32, 37, 84, 66, 91, 10] and [51], (1.4), [145], Prinzip 3, p. 11; explicit versions are in [127], and related estimates in [58, 56, 65, 49, 120, 105]. We use it only to see that the number of zeros of all $L(s,\chi)$ modulo $q$, counted with multiplicity, with $\lambda\le3$ and $|\gamma| \le T$ is bounded independently of $q$, for $T=2$ and for $T=\mathcal{L}$: take $1-\sigma=3/\mathcal{L}$; then $(qT)^{(2+\varepsilon)(1-\sigma)}\ll(q\mathcal{L})^{7/\mathcal{L}}\ll1$.

Exceptional zeros.

Input 3.10 (Siegel zeros [50], Corollary 1, p. 406). Suppose $L(\beta_0,\chi)=0$ for a real character $\chi$ modulo $q$, not necessarily primitive, and a real $\beta_0 \ge1-1/(3\log q)$, and put $\eta_0=((1-\beta_0)\log q)^{-1}$. For every $\delta>0$ there is an effectively computable constant $\eta_{\mathrm{HB}}(\delta)$ such that $\eta_0 \ge\eta_{\mathrm{HB}}(\delta)$ implies $P(a,q)\le q^{3+\delta}$ for every $a$ coprime to $q$.

The zero-location results of Heath-Brown and Xylouris that we use are stated in Section 7, where they are applied, and the inputs for the small first-zero regime in Section 12.

Prime detection

Our goal is to make a nonnegative weighted sum over primes in a reduced residue class strictly positive. The explicit formula expresses this as a comparison between a main term and a sum over zeros. We first fix the weight, then state the precise zero-sum bound that will suffice.

The kernel.

Fix

and the sixteen step heights $b_0,\ldots,b_{15}$ of Table 3. Put

$i$	$b_i$	$i$	$b_i$	$i$	$b_i$	$i$	$b_i$
0	0.02318741	4	0.23237155	8	0.47877833	12	0.75172165
1	0.07159337	5	0.29081583	9	0.54505502	13	0.82236428
2	0.12265085	6	0.35147264	10	0.61279917	14	0.89340300
3	0.17627704	7	0.41418551	11	0.68177389	15	0.96451862

For the target exponent $L$ put $A=L-2T$; at $L=3.99$ we have $A=3.156342$. The prime weight is

which is supported in $[A,L]$. Its Laplace transform is

Numerically $\Psi(0)=\kappa\sum_i b_i=0.193642\ldots$ and $H_0=0.037497\ldots$.

Xylouris [145], following Heath-Brown [51], uses the triangular weights

for $L'>2K>0$; the transform of $h_{L',K}$ is $e^{-(L'-2K)z}\left((1-e^{-Kz})/z\right)^2$.

Lemma 4.1 (Decomposition into positive triangular weights). For the step function $\psi$ and shifted convolution $h$ defined above,

Every triangle in this sum has left endpoint $A+k\kappa\ge A$.

Proof. For $i,j\ge0$ the convolution $\mathbf{1}_{[i\kappa,(i+1)\kappa)}*\mathbf{1}_{[j\kappa,(j+1)\kappa)}$ is the tent function of height $\kappa$ supported on $[(i+j)\kappa,(i+j+2)\kappa]$, which is $h_{(i+j+2)\kappa,\kappa}$. Expanding $\psi*\psi$ and shifting by $A$ gives the formula. Each $c_k$ is a sum of products of positive numbers, and the sum over $i+j=k$ is nonempty for $0\le k\le30$. $\square$

The positivity criterion

The following is Xylouris’s form of the explicit formula for a finite combination of triangles [145]; it refines Heath-Brown’s Lemmas 13.1 and 13.2 [51].

Input 4.2 (Xylouris’s criterion). Let $h=\sum_i\alpha_i h_{L_i,K_i}$ be a finite real combination of triangles with $L_i>2K_i+3$ for every $i$, and let $H$ be its Laplace transform. For every $\varepsilon>0$ there are $C_0=C_0(\varepsilon)$ and $q_0=q_0(\varepsilon)$ such that for every $q\ge q_0$ and every $a$ coprime to $q$,

where $\sum_{\rho}^{\prime}$ runs over the zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$, with multiplicity, in the square $1-C_0/\mathcal{L}\le\beta\le1$, $|\gamma|\le C_0/\mathcal{L}$.

Xylouris states the extension to combinations without a separate proof, and it follows from Heath-Brown’s proof of his Lemma 13.2 [51]. That proof starts from the explicit formula [51], which is linear in the weight, and bounds the zeros outside the square by summing over rectangles $1-\frac{m+1}{\mathcal{L}}\le\beta\le1-\frac{m}{\mathcal{L}}$, $\frac{n}{2\mathcal{L}}\le|\gamma|\le\frac{n}{\mathcal{L}}$ with $\max(m,n)\ge R$, using a zero-density bound $\ll e^{3m}(1+n/\mathcal{L})^{3/2}$ for each rectangle and $|F((1-\rho)\mathcal{L})|\ll e^{-(L'-2K)m}n^{-2}$ for each zero. For a combination with positive coefficients $c_k$ one has $|H|\le\sum_k c_k|H_k|$, so the same estimate holds with $L'-2K$ replaced by the smallest left endpoint, which exceeds 3; and the zeros in the square contribute at most $\sum|H((1-\rho)\mathcal{L})|$.

In the normalized coordinates (2) the square is $\{0 \le\lambda\le C_{0},\, |\mu| \le C_{0}\}$. Enlarging it only adds nonnegative terms to the zero sum, so the conclusion holds for every square $\{\lambda\le C_{P},\, |\mu| \le C_{P}\}$ with $C_{P} \ge C_{0}$. We fix

which fixes the prime window $R_{P}=\{0\le\lambda\le C_{P},\,|\mu|\le C_{P}\}$ of Section 2. Since $C_{P}$ is fixed and $\log\log\mathcal{L}\to\infty$, for large $q$ every zero in $R_{P}$ lies in the rectangle $R(l)$ of Section 2.

Definition 4.3 (Normalized zero sum). For the kernel $H$ in (4) and the prime window fixed by (5), put

where each zero is counted with its multiplicity.

Proposition 4.4 (A zero-sum bound that guarantees a prime). Let $L>3+2T$. There is $q_{0}$ such that for every $q\ge q_{0}$ with $W(q)\le1-2\eta$ and every $a$ coprime to $q$ there is a prime $p$ with

Here $W(q)$ is formed from the kernel and prime window for this fixed $L$; the threshold $q_{0}$ is independent of $a$ and of the zero configuration of $q$.

Proof. By Lemma 4.1, $h$ is a finite positive combination of triangles $h_{L_{i},\kappa}$ with $L_{i}-2\kappa=A+k\kappa\ge A>3$. So Input 4.2 applies with $\varepsilon=\eta H_{0}$; it gives a $q_{0}$ such that for every $q\ge q_{0}$ with $W(q)\le1-2\eta$,

The sum is therefore nonempty. Since $h$ vanishes outside $(A,L)$, some prime $p\equiv a\pmod q$ satisfies $A<\log p/\mathcal{L}<L$. $\square$

At $L=3.99$ the condition $L>3+2T=3.833658$ holds. Two features of Input 4.2 matter below.

(i) Occurrence form. The zero sum charges each zero occurrence separately, through the absolute value of its own term. Every bound in Section 5 is a bound for a sum of such individual terms. In particular we never need the cancellation between zeros that a grouped form of the explicit formula would record.

(ii) Size of a term. Since $|\Psi(\lambda-i\mu)|\le\Psi(\lambda)$ for $\lambda\ge0$,

A zero at distance $\lambda$ from $\sigma=1$ costs at most $e^{-A\lambda}$ times $H_{0}$. The difficulty is entirely with the number of zeros close to $\sigma=1$.

If no nonprincipal $L(s,\chi)$ has a zero in $R(l)$, then for large $q$ it has none in $R_{P}$, and $W=0$. This is the first exterior regime (Section 12).

Zero costs

The zero sum $W$ of Section 4 is a sum over characters. This section bounds the contribution of a single character, in terms of an anchor (Section 2.4): a number $\lambda_{*}$ such that no zero of $L(s,\chi)$ near $s=1$ has parameter below $\lambda_{*}$, at which the explicit formula is evaluated. For most characters the anchor is the parameter of a representative zero (Definition 6.4); the first family needs a finer treatment. The argument is Xylouris’s proof of his Lemma 3.10 [145], pp. 35–36, which we reproduce in the localized form we need.

The envelope functions

The function $\mathcal{G}_{\phi}(\lambda)$ below bounds the total cost of one character. The auxiliary functions $B_{\phi}$, $\widetilde{h}$, and $C$ separate the common exponential decay from the correction for a distinguished zero. The parameter $\phi$ will be $\phi(\chi)$, or an upper bound for it.

For real $\lambda$ let $\psi_{\lambda}(u)=\psi(u)e^{-\lambda u}$ and

so that $f_{\lambda}(t)=0$ for $t\geq T$. Let $F_{\lambda}$ be its Laplace transform. For $0\leq\phi\leq\frac{1}{3}$, $\lambda\geq0$ and $p\geq a\geq0$ put

Lemma 5.1 (Properties of the envelope functions). With the functions just defined, the following hold. Parts (i)–(iii) hold for every real $\lambda$; in (iv)–(v), take $0\leq\phi\leq1/3$, $\lambda\geq0$, and $p\geq a\geq0$.

(i) $f_{\lambda}\geq0$, $f_{\lambda}$ is Lipschitz on $[0,T]$, and $f_{\lambda}(T)=0$.

(ii) $\operatorname{Re}F_{\lambda}(iy)=|\Psi(\lambda+iy)|^{2}$ for real $y$.

(iii) $\operatorname{Re}F_{\lambda}(z)\geq|\Psi(\lambda+z)|^{2}$ for $\operatorname{Re}z\geq0$. In particular $\operatorname{Re}F_{\lambda}(z)\geq0$ for $\operatorname{Re}z\geq0$.

(iv) $H_{0}B_{\phi}(\lambda)=F_{\lambda}(-\lambda)+\frac{\phi}{2}f_{\lambda}(0)$ and $H_{0}C(p,a)=F_{p}(a-p)$. Moreover $C(a,a)=\widetilde{h}(a)$.

(v) $B_{\phi}$, $\mathcal{G}_{\phi}$ and $\widetilde{h}$ are positive and decreasing on $[0,\infty)$, and so is $\mathcal{G}_{\phi}/w$, where $w$ is either far weight of Section 6.

Proof. (i) Nonnegativity is clear. For $0\leq t<t'\leq T$, $|f_{\lambda}(t)-f_{\lambda}(t')|\leq2\lVert\psi_{\lambda}\rVert_{\infty}\int|\psi_{\lambda}(u+t)-\psi_{\lambda}(u+t')|\,\mathrm{d}u$, which is $O(t'-t)$ because $\psi_{\lambda}$ has bounded variation.

(ii) Let $a_{\lambda}(t)=\int\psi_{\lambda}(u)\psi_{\lambda}(u+|t|)\,\mathrm{d}u$ for $t\in\mathbb{R}$; it is even and integrable, and its Fourier transform is $|\widehat{\psi}_{\lambda}(y)|^{2}=|\Psi(\lambda+iy)|^{2}$. Hence $\operatorname{Re}F_{\lambda}(iy)=\int_{0}^{\infty}f_{\lambda}(t)\cos(yt)\,\mathrm{d}t=\int_{\mathbb{R}}a_{\lambda}(t)e^{-iyt}\,\mathrm{d}t=|\Psi(\lambda+iy)|^{2}$.

(iii) Apply Input 3.2 to $F_{1}=F_{\lambda}$ and $F_{2}(z)=\Psi(\lambda+z)^{2}$. Both are entire. By (i), $F_{\lambda}(z)=z^{-1}(f_{\lambda}(0)+\int_{0}^{T}f_{\lambda}'(t)e^{-zt}\,\mathrm{d}t)$, so $|F_{\lambda}(z)|\leq(f_{\lambda}(0)+\lVert f_{\lambda}'\rVert_{1})/|z|$ for $\operatorname{Re}z\geq0$; similarly $|\Psi(\lambda+z)|\leq(\psi_{\lambda}(0+)+V(\psi_{\lambda}))/|z|$ with $V$ the total variation. So both tend to 0 uniformly, and on $\operatorname{Re}z=0$ we have equality by (ii).

(iv) Substitute $v=u+t$: $F_{\lambda}(-\lambda)=2\iint\psi(u)\psi(v)e^{-\lambda(u+v)}e^{\lambda(v-u)}\,\mathrm{d}v\,\mathrm{d}u$ over $0\leq u\leq v\leq T$, which is the first term of $H_{0}B_{\phi}$; and $f_{\lambda}(0)=2\int\psi^{2}e^{-2\lambda u}$. The same substitution gives $F_{p}(a-p)=2\int_{0}^{T}\psi(u)e^{-2pu}\int_{u}^{T}\psi(v)e^{-a(v-u)}\,\mathrm{d}v\,\mathrm{d}u$. Finally $H_{0}C(a,a)=2\iint_{u\leq v}\psi(u)\psi(v)e^{-a(u+v)}=\Psi(a)^{2}$.

(v) All integrands are positive and decrease in $\lambda$. For the last claim,

and $2t-A<0$ on $[u_{0},x]$ (Lemma 6.2). $\square$

The number $\mathcal{G}(\lambda)$ is the cost, in units of $H_{0}$, that the argument below assigns to a character whose zeros near $s=1$ all have parameter at least $\lambda$. For a real character the factor $\phi=\frac{1}{3}$ may be replaced by $\frac{1}{4}$ (Proposition 5.5).

Smoothing

Since $\psi$ is a step function, $f_{\lambda}$ has corners at the multiples of $\kappa$ and does not satisfy Condition 1. We therefore work with smooth approximations of $\psi$ in the auxiliary functions only; the prime weight $h$ and its transform $H$ are never changed. For $\theta\in(0,1)$ fix a function $\psi^{\theta} \in C^{\infty}(\mathbb{R})$ with support in $(0,T)$ such that

for instance a mollification of $\psi\mathbf{1}_{[\theta/4,T-\theta/4]}$. Let $\Psi^{\theta}$, $f_{\lambda}^{\theta}$, $F_{\lambda}^{\theta}$, $B_{\phi}^{\theta}$, $\tilde{h}^{\theta}$ and $C^{\theta}$ be defined as above with $\psi^{\theta}$ in place of $\psi$, and $H_{0}$ kept.

Lemma 5.2 (Uniform smooth approximation). Fix $\theta\in(0,1)$ and $\Lambda\ge1$, and choose $\psi^{\theta}$ as above.

(i) For every $\lambda\in[0,\Lambda]$, $f_{\lambda}^{\theta}$ satisfies Condition 1 and Condition 2 with $x_{0}=T$. The quantities $x_{0}$, $x_{0}^{-1}$, $\sup|f_{\lambda}^{\theta}|$ and $\sup_{(0,T)}(|(f_{\lambda}^{\theta})'|+|(f_{\lambda}^{\theta})''|)$ are bounded independently of $\lambda\in[0,\Lambda]$; the bounds may depend on $\theta$ and $\Lambda$. Lemma 5.1(i)–(iv) hold for $\psi^{\theta}$.

(ii) For $\operatorname{Re} z \ge0$, $|\Psi(z)^{2}-\Psi^{\theta}(z)^{2}| \le3\theta$.

(iii) As $\theta\to0$, $B_{\phi}^{\theta} \to B_{\phi}$, $\tilde{h}^{\theta} \to\tilde{h}$ and $C^{\theta} \to C$ uniformly on $\{0 \le a \le p \le\Lambda\}$ and $0 \le\phi\le\frac{1}{3}$.

Proof. (i) The function $(t,\lambda) \mapsto f_{\lambda}^{\theta}(t)$ is smooth on $\mathbb{R}^{2}$, and it vanishes for $t \ge T$ because $\psi^{\theta}$ is supported in a compact subset of $(0,T)$; this gives Condition 1 and the uniform bounds. The proof of Lemma 5.1 applies verbatim to $\psi^{\theta}$, and Condition 2 is Lemma 5.1(i) and (iii) for $\psi^{\theta}$. (ii) For $\operatorname{Re} z \ge0$ we have $|\Psi(z)| \le\lVert\psi\rVert_{1} \le T < \frac{1}{2}$, $|\Psi^{\theta}(z)| \le\lVert\psi\rVert_{1}+\theta$ and $|\Psi(z)-\Psi^{\theta}(z)| \le\theta$; hence $|\Psi^{2}-(\Psi^{\theta})^{2}| \le\theta(2T+\theta) \le3\theta$. (iii) Each quantity is a bilinear expression in $(\psi,\psi)$ with a kernel bounded by $1$ on $[0,T]^{2}$ for the parameters in question, and $\lVert\psi\rVert_{\infty},\lVert\psi^{\theta}\rVert_{\infty} \le1$. □

The disc and a uniform zero count

Throughout this section $\lambda_{11}=0.05$, the smoothing parameter $\theta$ is the one fixed in stage (S3) of Section 13.1, and $\varepsilon_{1}=\eta/23$. Let $\mathcal{A}\subset[\lambda_{11},C_{P}]$ be the finite set consisting of the points of the grid $\lambda_{11}+h_{\mathcal{A}}\mathbb{Z}$ in $[\lambda_{11},C_{P}]$ together with the anchors of the first-family bounds of all leaves (Sections 5.6 and 5.7). For a leaf with first-zero cell $[a,b]$ and lower bound $p^{*}$ for $\lambda'$ (the lower end of its gap if it has one, and $p$ otherwise; Section 2.4), these are $a$ and $p^{*}$ if the leaf is inside and $p^{\circ}=\max(a,p^{*})$ if it is outside (Section 11.5); the mesh $h_{\mathcal{A}}>0$ is chosen so that $|B_{\phi}(x)-B_{\phi}(y)|\le\varepsilon_{1}/4$ for $\phi\in\{\frac{1}{4},\frac{1}{3}\}$ and $x,y\in[0,C_{P}]$ with $|x-y|\le h_{\mathcal{A}}$. In stage (S5) we fix a radius $\delta_{0}\in(0,\frac{1}{3}]$ such that Input 3.4 holds with radius $\delta_{0}$ for each of the finitely many pairs of a test function and a tolerance used in this paper: the function $f^{(1)}$ below with tolerance $1$, the functions $f_{\sigma}^{\theta}$ $(\sigma\in\mathcal{A})$ with tolerance $H_{0}\varepsilon_{1}/8$, and the detectors (Definition 8.1) of the near rows with the tolerances of Lemma 8.7. This is possible because Input 3.4 remains true for every smaller radius (see the remark after it). The disc of a character $\chi$ is the set of zeros $\rho$ of $L(s,\chi)$ with $|1-\rho|\le\delta_{0}$. For fixed $C$ and large $q$ the square $\{\lambda\le C,|\mu|\le C\}$ lies in the disc, since there $|1-\rho|\le\sqrt{2}C/\mathcal{L}$.

Lemma 5.3 (A uniform bound for zeros in a fixed window). Let $C\ge1$. There are $N=N(C)$ and $q_{0}$ such that for $q\ge q_{0}$ the following holds. If $\chi\ne\chi_{0}$ and every zero in the disc of $\chi$ has parameter at least $\lambda_{11}$, then $L(s,\chi)$ has at most $N$ zeros, counted with multiplicity, with $\lambda\le C$ and $|\mu|\le C$.

Proof. Let $K_{1}=1/(4C)$ and $\vartheta(z)=(1-e^{-K_{1}z})/z$, the transform of $\mathbf{1}_{[0,K_{1}]}$. Let $f^{(1)}(t)=2\int_{0}^{K_{1}-t}e^{-\lambda_{11}(2u+t)}\,\mathrm{d}u$ for $0\le t\le K_{1}$, and $f^{(1)}(t)=0$ for $t\ge K_{1}$. This is the function $f_{\lambda_{11}}$ of Section 5.1 for the step $\mathbf{1}_{[0,K_{1}]}$; it equals $(e^{-\lambda_{11}t}-e^{-\lambda_{11}(2K_{1}-t)})/\lambda_{11}$ on $[0,K_{1}]$, so it satisfies Condition 1, and $f^{(1)}(0)>0$. As in Lemma 5.1(iii), $\operatorname{Re}F^{(1)}(z)\ge|\vartheta(\lambda_{11}+z)|^{2}\ge0$ for $\operatorname{Re}z\ge0$.

Apply Input 3.4 to $f^{(1)}$ at $s=1-\lambda_{11}/\mathcal{L}$, with tolerance $1$ and a radius at most $\delta_{0}$. For large $q$, the fixed normalized square lies in this smaller disc. All its zeros still have parameter at least $\lambda_{11}$. In the remainder of this proof, “the disc” refers to this smaller disc. The point $s$ lies in the range (3.1) for large $q$, and $(s-\rho)\mathcal{L}=(\lambda_{\rho}-\lambda_{11})-i\mu_{\rho}$. Since $-\operatorname{Re}\chi(n)\leq\chi_{0}(n)$ for every $n$ and $f^{(1)}\geq0$, Input 3.3 gives

Hence $\sum_{\rho\ \mathrm{in\ the\ disc}}\operatorname{Re}F^{(1)}((\lambda_{\rho}-\lambda_{11})-i\mu_{\rho})\leq F^{(1)}(-\lambda_{11})+\frac{1}{6}f^{(1)}(0)+2=:c_{1}$ for large $q$. Every term is nonnegative, because $\lambda_{\rho}\geq\lambda_{11}$. For a zero with $\lambda\leq C$, $|\mu|\leq C$ put $z=\lambda-i\mu$; then $|K_{1}z|\leq\sqrt{2}/4<\frac{1}{2}$ and $|1-e^{-w}|\geq|w|-|w|^{2}e^{|w|}/2\geq|w|/2$ for $|w|\leq\frac{1}{2}$, so the term is at least $|\vartheta(z)|^{2}\geq K_{1}^{2}/4$. The number of such zeros is therefore at most $N=\lfloor4c_{1}/K_{1}^{2}\rfloor$. $\square$

The anchored envelope

The following inequality is the common source of all the costs below. For $\sigma\in\mathcal{A}$, a character $\chi\neq\chi_{0}$ and a zero $\rho$ put

By Lemma 5.1(iii), $\Phi_{\sigma}^{\theta}(\rho)\geq|\Psi^{\theta}(\lambda_{\rho}-i\mu_{\rho})|^{2}\geq0$ whenever $\lambda_{\rho}\geq\sigma$.

Lemma 5.4 (Anchored inequality). Fix the smoothing, finite anchor set $\mathcal{A}$, and common disc radius as in Section 5.3. There is $q_{0}$, uniform over $\chi\neq\chi_{0}$ and $\sigma\in\mathcal{A}$, such that for $q\geq q_{0}$,

Proof. Apply Input 3.4 to $f=f_{\sigma}^{\theta}$ at $s=1-\sigma/\mathcal{L}$ (so $t=0$), with tolerance $H_{0}\varepsilon_{1}/8$ and radius $\delta_{0}$, and bound the prime sum as in the proof of Lemma 5.3, using Input 3.3:

By Lemma 5.1(iv) for $\psi^{\theta}$ the right side is at most $\mathcal{L}(H_{0}B_{\varphi(\chi)}^{\theta}(\sigma)+\frac{1}{4}H_{0}\varepsilon_{1})$ for large $q$. Only the finitely many functions $f_{\sigma}^{\theta}$, $\sigma\in\mathcal{A}$, occur, so one $q_{0}$ serves them all. $\square$

Proposition 5.5 (Localized envelope). For $q\geq q_{0}$ the following holds. Let $\chi\neq\chi_{0}$ and $\lambda_{*}\in[\lambda_{11},C_{P}]$, and suppose that every zero in the disc of $\chi$ has parameter at least $\lambda_{*}$. Then

the sum running over the zeros of $L(s,\chi)$ in $R_{P}$ with multiplicity. In particular the left side is at most $G(\lambda_{*})+\varepsilon_{1}e^{-A\lambda_{*}}$, and at most $G_{1/4}(\lambda_{*})+\varepsilon_{1}e^{-A\lambda_{*}}$ if $\chi$ is real.

Proof. Let $N=N(C_{P})$ be as in Lemma 5.3. The parameter $\theta$ of stage (S3) satisfies

the supremum over $[0,C_{P}]$ and $\phi\in\{\frac{1}{4},\frac{1}{3}\}$ (Lemma 5.2(iii)). Let $\sigma\in\mathcal{A}$ be the largest grid point with $\sigma\leq\lambda_{*}$, so that $\lambda_{*}-\sigma\leq h_{\mathcal{A}}$. Let $\rho\in R_{P}$. Then $\rho$ is in the disc, so $\lambda_{\rho}\geq\lambda_{*}\geq\sigma$, and by (4.3), Lemma 5.2(ii) and Lemma 5.1(iii) for $\psi^{\theta}$,

The zeros of the disc that are not in $R_{P}$ have $\Phi_{\sigma}^{\theta}\geq0$. Summing over the at most $N$ zeros in $R_{P}$ and using Lemma 5.4 and (7),

and $B_{\varphi(\chi)}(\sigma) \le B_{\varphi(\chi)}(\lambda_*) + \frac{1}{4}\varepsilon_1$ by the choice of $h_{\mathcal A}$. Finally $\varphi(\chi) \le\frac{1}{3}$ always, $\varphi(\chi) = \frac{1}{4}$ for real $\chi$ (whose order is $2 \le\mathcal L$), and $B_\phi$ increases with $\phi$. □

Remark 5.6. Proposition 5.5 is a localized form of Xylouris’s Lemma 3.10 [145], pp. 35–36, applied with $m=1$, $f_1=f_\sigma^\theta$ and $H_2=(\Psi^\theta)^2$, for which his condition (3.59) holds with equality by Lemma 5.1(ii); the prime weight $H$ is then compared with $(\Psi^\theta)^2$ through Lemma 5.2(ii). We require only that the zeros in the disc have parameter at least $\lambda_*$, where Xylouris requires that $L(s,\chi)$ have no zero with $1-\lambda_*/\mathcal L<\beta\le1$ and $|\gamma|\le1$; his proof uses this hypothesis only for the zeros in the disc, which have $|\gamma|\le\delta_0<1$. His printed statement allows $f_1$ to be a sum of $m$ pieces; for $m>1$ his proof needs in addition that $f_{1i}(0)\ge0$ for each piece, and that the sum over the square be enlarged to a sum over a disc common to all pieces before Input 3.4 is applied to them. With $m=1$ neither point arises. Finally, we use finitely many anchors, so that no uniformity in the anchor is needed. The coefficient $\varphi(\chi)/2$ comes from Input 3.4; the printed Lemma 3.10 uses $\varphi(\chi)\le\frac{1}{3}$.

Ordinary and reserved characters

Let $\chi$ be a nonprincipal character (in the case tree, one outside the first family; in Theorem 12.6, any), and let $\lambda_\chi$ be the parameter of its $T^*$-representative or of its height-one representative (Definition 6.4). If $L(s,\chi)$ has a zero in $R_P$, then that zero lies in $R(l)$ and has $|\gamma|\le C_P/\mathcal L<T^*$, so the representative exists and $\lambda_\chi\le C_P$. In the regimes in which we use these costs $\lambda_1\ge0.1$, so $\lambda_\chi\ge\lambda_{11}$.

Corollary 5.7 (Cost of a character with a representative). Suppose $\lambda_1\ge0.1$. Let $\chi\ne\chi_0$ have a zero in $R_P$, and let $\lambda_\chi$ be the parameter of either its $T^*$-representative or its height-one representative. For sufficiently large $q$, its contribution to $W$ is at most

If $\chi$ is real, $G$ may be replaced by $G_{1/4}$. The threshold is uniform over these choices of character and representative.

Proof. We check the hypothesis of Proposition 5.5 with $\lambda_*=\lambda_\chi$. A zero in the disc with parameter at most $C_P$ lies in $R(l)$ for large $q$, since $C_P<\frac{1}{3}\log\log\mathcal L$ and $|\gamma|\le\delta_0<1\le l$; it has $|\gamma|\le\delta_0<\frac{1}{2}\le T^*$, so its parameter is at least $\lambda_\chi$ by the minimality of the representative. A zero in the disc with parameter above $C_P$ has parameter above $\lambda_\chi$. □

The first family inside the buffer

Suppose that $\rho_1\in R_B$, that $\lambda_1\in[a,b]$ with $a\ge\lambda_{11}$, and that every zero occurrence of the first family in $R(l)$ other than the distinguished ones has parameter at least $p$, where $p\le C_P$; equivalently, $p\le\lambda'$. We assume that $a,p\in\mathcal A$ (in the leaves they are the numbers $a$ and $p^*$ of Section 5.3). The distinguished occurrences and their number $\alpha$ per character are those of Definition 2.2 and the table after it. Let $\phi_1=\frac{1}{4}$ if $\chi_1$ is real and $\phi_1=\frac{1}{3}$ otherwise, and put

Proposition 5.8 (Cost of the first family inside the buffer). Suppose $\rho_1\in R_B$, $\lambda_1\in[a,b]$, $a,p\in\mathcal A$, $a\ge\lambda_{11}$, and $p\le\min(\lambda',C_P)$. For sufficiently large $q$, the contribution of all $n$ characters of the first family to $W$ is at most $J_{\mathrm{old}}+n\varepsilon_1$. If also $p\ge b$, it is at most $J_{\mathrm{new}}+n\varepsilon_1$. Here $J_{\mathrm{old}}$ and $J_{\mathrm{new}}$ are the displayed functions of $a,p,n,\alpha$, and $\phi_1$.

The two bounds use different anchors. The bound $J_{\mathrm{old}}$ uses the lower endpoint $a$ of the first-zero cell. The bound $J_{\mathrm{new}}$ uses $p$, the lower bound for the remaining zeros, and estimates the distinguished zero together with its correction in the explicit formula. Their combined expression is smaller than a separate estimate of the two terms.

Proof. It suffices to treat one first-family character $\chi$; the bound for its conjugate is the same, because $R_P$ and the disc are symmetric under conjugation. A zero of the disc of $\chi$ lies in $R(l)$ or has parameter above $\frac{1}{3}\log\log\mathcal{L}>C_P$. So every zero of the disc has parameter at least $\lambda_1\geq a$, and every non-distinguished one has parameter at least $p$. Write $S$ for the distinguished occurrences of $\chi$; they lie in $R_B$, hence in the disc.

The bound $J_{\mathrm{old}}$. Every non-distinguished occurrence has parameter at least $\max(p,\lambda_1)\geq\max(p,a)$, and $J_{\mathrm{old}}$ decreases as $p$ increases; so we may and do assume $p\geq a$. For a non-distinguished $\rho\in R_P$ we then have $\lambda_\rho\geq p\geq a$, and $|H((1-\rho)\mathcal{L})|\leq e^{-Ap}|\Psi(\lambda_\rho-i\mu_\rho)|^2\leq e^{-Ap}(\Phi_a^\theta(\rho)+3\theta)$. By Lemma 5.4 at $\sigma=a$, since every term $\Phi_a^\theta$ in the disc is nonnegative and there are at most $N$ zeros in $R_P$,

For $\rho\in S$ put $X_\rho=|\Psi(\lambda_1-i\mu_\rho)|^2$; then $\Phi_a^\theta(\rho)\geq X_\rho-3\theta$ and $|H((1-\rho)\mathcal{L})|=e^{-A\lambda_1}X_\rho$. We add $e^{-A\lambda_1}X_\rho$ for every $\rho\in S$; for those in $R_P$ this is their contribution, and for the others it is a nonnegative addition. By (5.1) the total is at most

If $\lambda_1\leq p$, each summand increases with $X_\rho$, and $X_\rho\leq\Psi(\lambda_1)^2\leq\Psi(a)^2=H_0\widetilde{h}(a)$; so the total is at most $H_0[e^{-Ap}(B_{\phi_1}^\theta(a)-\alpha\widetilde{h}(a))+\alpha e^{-A\lambda_1}\widetilde{h}(a)+\varepsilon_1]$, which is at most $H_0(J_{\mathrm{old}}/n+\varepsilon_1)$. If $\lambda_1>p$, each summand is $\leq0$, and the total is at most $H_0[e^{-Ap}B_{\phi_1}^\theta(a)+\varepsilon_1]\leq H_0[e^{-Ap}(B_{\phi_1}^\theta(a)-\alpha\widetilde{h}(a))_++\alpha e^{-Ap}\widetilde{h}(a)+\varepsilon_1]$, which is again at most $H_0(J_{\mathrm{old}}/n+\varepsilon_1)$ because $p\geq a$.

The bound $J_{\mathrm{new}}$. Now $p\geq b\geq\lambda_1$. Apply Lemma 5.4 at $\sigma=p$. The non-distinguished zeros of the disc have $\Phi_p^\theta\geq0$, and for those in $R_P$, $|H((1-\rho)\mathcal{L})|\leq e^{-Ap}(\Phi_p^\theta(\rho)+3\theta)$. Therefore the contribution of $\chi$ is at most

The indicator may be replaced by $1$, since the term it multiplies is nonnegative, and $|\Psi(\lambda_1-i\mu_\rho)|^2\leq|\Psi^\theta(\lambda_1-i\mu_\rho)|^2+3\theta$. For a real number $y$ consider

where, by Lemma 5.1(ii) for $\psi^\theta$ and the definition of $f_p^\theta$,

Since $\lambda_1\leq p$, the integrand is nonnegative, so $k\geq0$ and $D(y)\leq D(0)=\int_0^\infty k(t)\,\mathrm{d}t$. Moreover $k(t)$ decreases as $\lambda_1$ increases, for each $t$, because both $e^{-\lambda_1t}$ and the bracket decrease. Hence

by Lemma 5.1(iv). By Lemma 5.2(ii),(iii), $H_0|\widetilde{h}^\theta(a)-\widetilde{h}(a)|\leq3\theta$ and $H_0|C^\theta(p,a)-C(p,a)|\leq3\theta$. So each of the $\alpha\leq2$ distinguished occurrences contributes at most $H_0(e^{-Aa}\widetilde{h}(a)-e^{-Ap}C(p,a))+9\theta$, and with (5.1) the contribution of $\chi$ is at most $H_0(J_{\mathrm{new}}/n+\varepsilon_1)$. $\square$

We write $J_{1}=\min(J_{\mathrm{old}},J_{\mathrm{new}})$, where $J_{\mathrm{new}}$ is omitted when $p<b$. In every leaf of the case tree the anchor $p$ is at most $2.293<3\le C_{P}$, so the hypothesis $p\le C_{P}$ holds. In $J_{\mathrm{new}}$ the quantity $B_{\phi_{1}}(p)-\alpha C(p,a)$ may be negative; the argument does not require it to be positive.

The first family outside the buffer

Suppose now that $\rho_{1}\notin R_{B}$. Since $\lambda_{1}<C_{B}$ in the regimes where this occurs, this means $|\mu_{1}|>C_{B}$. The global zero $\rho_{1}$ is not in $R_{P}$ and contributes nothing to $W$, but the first-family characters may have other zeros in $R_{P}$. As before, the distinguished occurrences are $\rho_{1}$ (and $\overline{\rho_{1}}$) for $\chi_{1}$ and $\overline{\rho_{1}}$ for $\overline{\chi_{1}}$, and every other zero occurrence of the first family in $R(l)$ has parameter at least $p\ge a$, where $p\in A$ (in the leaves, $p=p^{\circ}$).

For a first-family character $\chi$ with a zero in $R_{P}$, let $t_{\chi}$ be the least parameter of its zeros in $R_{P}$. Then $t_{\chi}\ge p$, and the corresponding zero has $|\gamma|\le C_{P}/\mathcal{L}\le1$.

Proposition 5.9 (Cost of the first family outside the buffer). Suppose $\rho_{1}\notin R_{B}$, $\lambda_{1}\ge a\ge\lambda_{11}$, and $p\in A$ satisfies $a\le p\le\lambda'$. For a first-family character with a zero in $R_{P}$, let $t_{\chi}$ be the least parameter of its zeros in $R_{P}$. Let $\alpha$ be the number of distinguished occurrences per character, and put $\phi_{1}=1/4$ for a real $\chi_{1}$ and $\phi_{1}=1/3$ otherwise. There is a constant $c_{\mathrm{out}}$, depending only on the fixed choices of stages (S1)–(S3) of Section 13.1 (in particular on $\theta$), such that for $q\ge q_{0}$ the following holds. Each first-family character $\chi$ with a zero in $R_{P}$ contributes at most

to $W$, and a first-family character without a zero in $R_{P}$ contributes nothing.

Proof. Apply Lemma 5.4 at $\sigma=p\in A$. For $\rho\in R_{P}$ we have $\lambda_{\rho}\ge t_{\chi}\ge p$ and $|H((1-\rho)\mathcal{L})|\le e^{-At_{\chi}}(\Phi_{p}^{\theta}(\rho)+3\theta)$. The non-distinguished zeros of the disc have $\Phi_{p}^{\theta}\ge0$, and so does a distinguished occurrence with $\lambda_{\rho}\ge p$. A distinguished occurrence $\rho$ in the disc with $\lambda_{\rho}<p$ has $\lambda_{\rho}-p\in[a-p,0)\subseteq[-C_{P},0]$ and $|\mu_{\rho}|>C_{B}$. Since $f_{p}^{\theta}$ is smooth on $[0,T]$ and vanishes near $T$, two integrations by parts give, for $x\in[-C_{P},0]$ and real $Y\ne0$,

where $c_{\mathrm{out}}=C_{P}f_{p}^{\theta}(0)+|(f_{p}^{\theta})'(0)|+e^{C_{P}T}\|(f_{p}^{\theta})''\|_{1}$, maximized over the finitely many anchors $p$ in use; the first term is bounded by $f_{p}^{\theta}(0)|x|/Y^{2}$. So the distinguished terms subtracted in Lemma 5.4 total at least $-\alpha c_{\mathrm{out}}/C_{B}^{2}$, and summing as in Proposition 5.5, with (5.1), gives the claim. $\square$

In the leaf programs these contributions are the hidden columns: at most one per first-family character, carrying the value $t_{\chi}$, the objective $e^{-At_{\chi}}B_{\phi_{1}}(p)$ and, because the zero realizing $t_{\chi}$ has $|\gamma|\le1$, the far weight $w(t_{\chi})$ (Section 11.5).

The objective and the error allowance

In a leaf of the case tree (Section 2.4) the unknown characters are described as follows (Section 11.5): the first family, possibly a reserved second family of $n_{2}\in\{1,2\}$ characters with height-one representatives in a given interval $[l_{02},h_{2}]$, and the ordinary characters, described by their $T^{*}$-representatives. By Corollary 5.7 and Propositions 5.8 and 5.9,

where $\phi_{2}=\frac{1}{4}$ if the reserved family is a single real character and $\phi_{2}=\frac{1}{3}$ otherwise, $\nu_{\chi}$ is the parameter of the height-one representative of a reserved character. The first-family term $J_{1}$ is $J_{1}$ for an inside leaf and $\sum_{\chi}e^{-At_{\chi}}B_{\phi_{1}}(p)$ for an outside leaf. In the latter case it is represented by hidden columns, rather than by the constant part of the leaf objective (Section 11.5).

There are two sources of error in (8). The envelope errors total less than $19\varepsilon_{1}$ for the ordinary characters, by (10), and at most $4\varepsilon_{1}$ for the first and reserved families. The outside first-family bound has the additional error $4c_{\mathrm{out}}/(H_{0}C_{B}^{2})$. We choose

so $E\leq2\eta$. The error $\eta$ in the prime-detection criterion is accounted for separately in Proposition 4.4.

Every leaf objective includes an allowance of $5\eta$. Consequently its value bounds $W+3\eta$, and therefore also $W+2\eta$, as required by the positivity criterion. These choices are made uniformly over the finite collection of leaves in Section 13.1.

Far density and representatives

The per-character costs must be summed over many characters. A weighted density estimate supplies a common budget for that sum. Its zeros have physical height at most 1, so we also specify how to choose representatives within this height range.

Xylouris’s weighted density lemma

The following is Xylouris’s improvement [145], Lemma 5.1, (5.19), p. 66 of Heath-Brown’s Lemma 11.1 [51]; the latter is the case $J=1$, $w_{0}\equiv1$. (Xylouris writes $M$ for our $J$; we reserve $M$ for a separation constant.)

Input 6.1 (Far density). Let $\varepsilon,c_{1},c_{2}>0$, $J\in\mathbb{N}$, and $\alpha_{1},\ldots,\alpha_{J}\geq0$ with $\sum_{i}\alpha_{i}=1$. Put

and let $w_{0}:[u_{0},x]\to\mathbb{R}$ be continuous and continuously differentiable except at finitely many points, with $1\ll w_{0}\ll1$ and $w_{0}'\ll1$. Choose for each nonprincipal character $\chi$ modulo $q$ at most one zero $\rho_{\chi}$ of $L(s,\chi)$ with $|\operatorname{Im}\rho_{\chi}|\leq1$ and $\lambda_{\chi}\leq\lambda_{0}=\frac{1}{3}\log\log\mathcal{L}$. Then for $q\geq q_{0}$, with $q_{0}$ depending on all the parameters,

In [145] the statement is made for the zeros $\rho^{(k)}$ that are chosen for the characters counted by $N(\lambda_{0})$, one zero per character [145], §3.2.2, pp. 25–26, [51], §11; any such choice is allowed, and the estimates in the proof [145], pp. 67–72 are uniform over these choices. This uniformity is needed because the selected zeros, including the $T^{*}(q)$-representatives below, depend on $q$. A nonreal character and its conjugate are different characters, and each contributes its own term; the principal character is excluded, since $L(s,\chi_{0})$ has no zero in the relevant region for large $q$ [145], p. 21. The hypotheses on $w_{0}$ are satisfied by our profiles, which are those of Xylouris’s own application [145], (6.20)–(6.21), p. 80.

The two weight profiles

We use two profiles of the shape [145], (6.20), both with $J=10$ and $\varepsilon_{0}=10^{-7}$:

Profile I has $c_{1}=0.09035$, $c_{2}=0.235968$ and $\theta_{\mathrm{far}}=1.28683$, and profile II has $c_{1}=0.0821922$, $c_{2}=0.2170903$ and $\theta_{\mathrm{far}}=1.4964274$. The weights $\alpha_{i}$ are the exact decimals of Table 4; in each profile they sum to 1. Both profiles satisfy the hypotheses of Input 6.1: $w_{0}$ is continuous and positive on $[u_{0},x]$, and it is smooth except at $t=u_{0}+c_{2}$.

$i$	profile I $\alpha_i$	profile II $\alpha_i$
1	0.0788827218	0.0806195583
2	0.0849386148	0.0862705300
3	0.0895629779	0.0905489307
4	0.0938231516	0.0944644867
5	0.0979710491	0.0982543287
6	0.1021284916	0.1020315591
7	0.1063732939	0.1058668825
8	0.1107651869	0.1098131140
9	0.1153565492	0.1139152197
10	0.1201979632	0.1182153903

Put

a positive decreasing function of $\lambda$. Let $V$ be the right-hand side of Input 6.1 with $\varepsilon=0$. Rigorous enclosures give

Taking $\varepsilon=\eta J^2$ in Input 6.1, we obtain for $q\geq q_0$ and every choice of zeros as in Input 6.1:

Each leaf of the case tree uses one of the two profiles, recorded in its data. Both are fixed finite choices within Input 6.1.

Lemma 6.2 (Comparison of the prime weight and the far weight). For both profiles, $x\leq1.1737$. Hence $A>2x$ for $L=3.99$, and for every $\lambda\geq0$ the function $\lambda\mapsto e^{-A\lambda}/w(\lambda)$ is decreasing. In particular $e^{-A\lambda}\leq w(\lambda)/w(0)$.

Proof. The values are $x=1.1736846\ldots$ and $x=1.1303335\ldots$, so $2x<2.348<A=3.156342$. Now $e^{-A\lambda}/w(\lambda)=\int_{u_0}^{x}w_0(t)^2e^{(2t-A)\lambda}\,\mathrm{d}t$, and $2t-A<0$ on $[u_0,x]$. $\square$

With $w(0)=10.7054\ldots$ (profile I) and $w(0)=13.1730\ldots$ (profile II), Lemma 6.2 and (9) give, for any selection satisfying Input 6.1,

This bound controls errors that are summed over unboundedly many characters.

Representatives

The leaf programs describe an ordinary character through one parameter, the parameter of a representative zero. For the near-density argument of Section 8 the representative must be separated in height from any zero with a smaller parameter, and we arrange this by choosing the height of the strip in which representatives are taken.

Fix $s_{\max}=3$; every anchor used in the near rows is at most $2.3$ (Section 9), and every ordinary entry uses an anchor at most $s_{\max}$. Let $K_0$ be an upper bound for the number of zeros, of all nonprincipal characters modulo $q$ and counted with multiplicity, with $\lambda\leq s_{\max}$ and $|\gamma|\leq2$. By a log-free zero-density estimate (Input 3.9), $K_0$ may be taken independent of $q$. Next fix $M$ as in Section 13; it depends on $K_0$.

Lemma 6.3 (A strip boundary separated from low-parameter zeros). Let $K_{0}$ bound the number of zeros with $\lambda\le s_{\max}=3$ and $|\gamma|\le2$, as above, and fix $M>0$. If $\mathcal{L}>4M(K_{0}+1)$, there is $T^{*}\in[1/2,1]$ such that no nonprincipal $L(s,\chi)$ has a zero with $\lambda\le s_{\max}$ and $\bigl||\mu|-T^{*}\mathcal{L}\bigr|<M$. We take $T^{*}=T^{*}(q)$ to be the least such number.

Proof. The at most $K_{0}$ zeros in question exclude at most $K_{0}$ open intervals of $T$ of length $2M/\mathcal{L}$ each. The interval $[1/2,1]$ has length $1/2>2MK_{0}/\mathcal{L}$, so the remaining set is a nonempty finite union of closed intervals, and it has a least element. $\square$

Definition 6.4 (Representatives). For a nonprincipal character $\chi$, its $T^{*}$-representative is a zero of $L(s,\chi)$ in $R(\ell)$ with $|\gamma|\le T^{*}$ and least parameter, if there is one; its parameter is $\lambda_{\chi}$. Representatives are used for the characters outside the first family; in Theorem 12.6 all characters use their height-one representatives. The height-one representative is defined in the same way with the bound $|\gamma|\le1$. The reserved family of a leaf (Section 2.4 and Definition 11.2) and an inside first family use height-one representatives.

Every representative has $|\gamma|\le1$, so the far bound (9) applies to any selection of representatives, one per character. Representatives for conjugate characters may be chosen as conjugate zeros, with the same parameter, because every selection region is symmetric in height. The $T^{*}$-representative of $\chi$ satisfies $\lambda_{\chi}\ge$ the parameter of its height-one representative. Shrinking the strip can only increase the minimum parameter; in terms of real parts, the selected zero can only move to the left.

The far constraint of a leaf

In a leaf program (Section 2.4), the characters other than the first family are placed in bins by the parameters of their representatives, and each bin has a column (Sections 10 and 11). The column of a bin $[\mathrm{lo},\mathrm{hi}]$ has far weight $W_{c}=\lfloor Sw(\mathrm{hi})\rfloor$; since $w$ is decreasing, each character in the bin contributes at least $w(\mathrm{hi})$ to the left side of (9). The tail column collects the ordinary characters $\chi$ with a zero in $R_{P}$ and $\lambda_{\chi}\ge R$, where $R=\max(3,r)$ and $r$ is the leaf’s lower bound for ordinary representatives. Its value is the mass $x_{\mathrm{tail}}=\sum w(\lambda_{\chi})$ over these characters, with far weight $S$. The far budget of the leaf is

Here the last term is present for an inside leaf, that is when $\rho_{1}\in R_{B}$. Then $|\gamma_{1}|\le C_{B}/\mathcal{L}\le1$, so $\rho_{1}$ and $\overline{\rho_{1}}$ are the height-one representatives of $\chi_{1}$ and $\overline{\chi_{1}}$, and they contribute $nw(\lambda_{1})\ge nw(b)$ to (9). A reserved second family is charged either through its own columns, which carry its far weight, or by subtracting the fixed amount $n_{2}\lfloor Sw(\mathrm{hi}_{2})\rfloor$ from $F$ (Section 11.5).

Zero location

This section collects the lower bounds for $\lambda_{1}$, $\lambda'$, $\lambda_{2}$ and $\lambda_{3}$ that the case tree uses. Most are printed results of Heath-Brown [51] and Xylouris [145]; we state each in the form used, with its scope. The others are new consequences of the same explicit formula, proved here by the positivity method; one of them uses a sharpened conductor coefficient, which we prove in Section 7.6. None of the results of this section depends on the exponent $L$.

Conventions

We fix once and for all an admissible height function $l(q)$ as in Input 3.5, for instance the least one. All the zero-location results we quote are proved for an arbitrary integer $l$ with $1\le l\le\mathcal{L}/10$ for which $R(10l)\setminus R(l)$ contains no zero; their proofs use $l$ only through these two properties, and their thresholds depend only on fixed data. So they hold simultaneously for our $l(q)$.

The zeros $\rho_{1},\rho_{2},\rho_{3}$ are selected as in [145], (3.11)–(3.12), p. 21 (which follows [51], §6). At step $k$, remove the characters in the families $\mathcal{F}_{1},\ldots,\mathcal{F}_{k-1}$ and choose a zero of maximal real part in $R(l)$ among the remaining nonprincipal $L$-functions. Each real character is removed once. This is the successive-minimum convention of Definition 2.1. The additional zero $\rho'$ of the first family is selected as in [145]: $\rho'=\rho_{1}$ if $\rho_{1}$ is multiple, and otherwise $\rho'$ has maximal real part among the zeros of $L(s,\chi_{1})$ in $R(l)$ other than $\rho_{1}$ (and other than $\overline{\rho}_{1}$ if $\chi_{1}$ is real). Its parameter is the $\lambda'$ of Section 2 (Heath-Brown also writes $\lambda'$; Xylouris indexes this zero by 0). When several zeros qualify (ties in the real part, or several characters attaining a minimum), any of them may be selected: the proofs of the quoted results use only the extremal property of the selection, so they hold for every such choice. This is why the configuration sets of Definition 11.2 quantify over the admissible choices. All the results of this section concern zeros in $R(l)$ at any height up to $l$, so they apply whether $\rho_{1}$ is inside or outside the buffer.

Printed tables. Every printed bound is used as an exact implication for $q\geq q_{0}$. Each of them was obtained by showing that a certain inequality fails with a positive margin, which persists when the error term $\varepsilon$ is small enough [145]; for Heath-Brown’s Tables 2 and 3 the printed values lie a little below the computed roots [51]. Heath-Brown states no convention for his Tables 4 and 7, and we have recomputed their roots from his printed parameters; every recomputed root exceeds the printed value, and each printed value is the root rounded down. The closest case in Table 4 is row 1.05, with root 1.43904 against 1.439, and the closest in Table 7 are rows 0.20 and 0.55, with 2.01015 against 2.01 and 1.00036 against 1.00. The rows of Table 4 are derived under the side conditions [51]; we have checked from the printed parameters that these hold for every row from 0.35 on (the tightest is (8.6), with slacks $3.8\cdot10^{-6}$, $3.8\cdot10^{-6}$ and $2.3\cdot10^{-6}$ for the rows 1.05, 1.15 and 1.294). For the row $(0.3,2.293)$ they need not hold, but that row also follows from Heath-Brown’s Table 3, which gives $\lambda'\geq2.43$ for $\lambda_{1}\leq0.30$ [51]. These checks, and the recomputation of the roots, were made in floating point, separately from the replay. A separate program compares every printed number that the proof uses, in the code and in this paper, with the text of the source pages and with the printed scope of its row; each agrees with the printed value or is weaker than it in the safe direction. Results stated in the form $c-\varepsilon$ are used with an explicit reduction: $0.857<\frac{6}{7}$ in Input 7.4, and $1.09<\frac{12}{11}$ and $\varepsilon=0.001$ in Section 12.

Types and the first zero.

Proposition 7.1 (Lower bounds for a nonreal first zero). For all sufficiently large $q$, the first family satisfies

Thus $\lambda_{1}\leq0.44$ forces both $\chi_{1}$ and $\rho_{1}$ to be real. These bounds are [145].

Xylouris’s lemma gives $\lambda_{1}>0.440$, 0.493, 0.478, 0.498 and 0.628 for characters of order $\geq6$, $=5$, $=4$, $=3$ and $=2$ respectively; the minimum over orders at least 3 is 0.440.

The test function.

The location inequalities use one family of test functions. We define it here so that all parameters in the statements below are specified before they occur. For $\gamma>0$, put

The function $f_{\gamma}$ is the autocorrelation of $x\mapsto(\gamma^{2}-x^{2})_{+}$, the test function of [145]; it satisfies Conditions 1 and 2 (Lemma 9.1), and its transform $F$ is positive and decreasing on $\mathbb{R}$.

Printed zero-location inputs.

Input 7.2 (Real first zero). Let $\chi_{1}$ and $\rho_{1}$ be real. For $q \ge q_{0}$:

(a) [51], Tables 3 and 4, Lemmas 8.2 and 8.3. If $\lambda_{1} \le t$ then $\lambda' \ge c$, for the 24 pairs $(t,c)$ from $(0.3,2.293)$ to $(1.294,1.294)$ of the table. For a real $\rho'$ this follows from $\lambda' \ge2.427$ (Lemma 8.2).

(b) [51], Lemma 8.4. For every $\varepsilon> 0$: $\lambda' \ge(2-\varepsilon)\log\lambda_{1}^{-1}$ if $\lambda_{1} \le0.2$ and $q \ge q(\varepsilon)$; and $\lambda' \ge\max(\frac{3}{2}\log\lambda_{1}^{-1},1.294)$ for every $\lambda_{1}$.

(c) [51], Table 7 and Lemma 8.7. If $\lambda_{1} \le t$ then $\lambda_{2} \ge c$, for the 16 pairs $(t,c)$ from $(0.12,2.56)$ to $(0.745,0.745)$ of the table; these hold whether or not $\chi_{2}^{4}=\chi_{0}$ (the row $t=0.10$ is not used).

Entry	Anchor $\sigma$	Kept set $S$	Response bound $r$	Diagonal numerator	Rows
(a) ordinary $\chi$, bin $[\mathrm{lo},\mathrm{hi})$, at its $T^{*}$-representative	$\min(\mathrm{lo},s)$	$\{\rho_{\chi}\}$	$F(\mathrm{hi}-\sigma)-\frac{1}{6}f(0)$	$D(s-\sigma)-d_{1}$	all three
(b) inside first family: $\chi_{1}$ at $\gamma_{1}$, and $\overline{\chi}_{1}$ at $-\gamma_{1}$	$\min(a,s)$	$\{\rho_{1}\}$, resp. $\{\overline{\rho}_{1}\}$	$F(b-\sigma)-\frac{1}{6}f(0)$	$D(s-\sigma)-d_{1}$	family, graded
(c) type rc: $\chi_{1}$ at $\pm\gamma_{1}$	$s_{1}$	$\{\rho_{1},\overline{\rho}_{1}\}$	$F(b-s_{1})+R_{\mathrm{lo}}-\frac{1}{6}f(0)$	$D(0)-d_{1}+(c^{\mathrm{hi}}-d_{1})_{+}$	family
(d) second zero, type complex: $\chi_{1}$ at $\gamma_{1},\gamma'$, $\overline{\chi}_{1}$ at $-\gamma_{1},-\gamma'$	$s_{1}$	$\{\rho_{1},\rho'\}$ and conjugates	$F(b-s_{1})+R_{p}-\frac{1}{6}f(0)$ at $\gamma_{1}$; $F(\mathrm{hi}'-s_{1})+R_{q}-\frac{1}{6}f(0)$ at $\gamma'$	$D(0)-d_{1}+(c^{\mathrm{hi}}-d_{1})_{+}$	family
(e) shifted first family: $\chi_{1}$ at $\gamma_{1}$, and $\overline{\chi}_{1}$ at $-\gamma_{1}$	$s$	$\{\rho_{1}\}$; $\{\rho_{1},\overline{\rho}_{1}\}$ for rc	$F(b-s)-\mathbf{1}_{\mathrm{rc}}C_{Z}-\frac{1}{6}f(0)$	$D(0)-d_{1}$	shifted
(f) reserved family, at height-one representatives	$\sigma_{2}$	the representative	$F(\mathrm{hi}_{2}-\sigma_{2})-\frac{1}{6}f(0)$	$D(s-\sigma_{2})-d_{1}$	all three
(g) outside first family: $\chi_{1}$ at $\gamma_{1}$ (and $\overline{\chi}_{1}$), and one hidden entry per character	$s_{1}$	$\{\rho_{1}\}$; $\{\rho_{h}\}$	$F(b-s_{1})-\frac{1}{6}f(0)$; $F(\mathrm{hi}_{h}-s_{1})-\frac{1}{6}f(0)$	$D(0)-d_{1}$	family

(d) [51], Lemma 8.8. $\lambda_{2} \ge\max((\frac{12}{11}-\varepsilon)\log\lambda_{1}^{-1},0.745)$ for $q \ge q(\varepsilon)$.

Heath-Brown remarks that the estimates of his §8 have not been proved for extremely small values of $\lambda_{1}$ [51], §8. We use (a)–(d) only for $\lambda_{1}$ bounded below by a fixed positive constant: $\lambda_{1} \ge0.1$ in the case tree, and, in Theorem 12.5, $\lambda_{1}$ at least the fixed constant $u_{\mathrm{exc}}>0$ of Section 12.

Input 7.3 (Nonreal first character or zero). Let $\chi_{1}$ or $\rho_{1}$ be nonreal. For $q \ge q_{0}$:

(a) [145], Table 2′, p. 62. If $\lambda_{1} \le t$ then $\lambda' > c$, for the pairs $(0.46,1.85),\ldots,(0.827,0.827)$ of the table.

(b) [145], Table 3, p. 45. If $\operatorname{ord}\chi_{1} \in\{2,3,4\}$ and $\lambda_{1} \le t$, then $\lambda' > c$, for the pairs $(0.38,2.53),\ldots,(1.099,1.099)$; in particular $\lambda_{1} \le0.86$ implies $\lambda' > 1.35$.

(c) [145], Tables 6 and 7, p. 53. If $\lambda_{1} \le t$ then $\lambda_{2}>c$; Table 6 concerns a real $\chi_{1}$ (Fall 7), Table 7 all cases.

(d) [51], Lemma 9.4 and Table 10. $\lambda_{2} \ge0.702$ always, and $\lambda_{1} \le0.70$ implies $\lambda_{2} \ge0.704$.

Input 7.4 (The third family). For $q \ge q_{0}$:

(a) [51], Lemma 10.3. $\lambda_{3} \ge\frac{6}{7}-\varepsilon$; we use $\lambda_{3}>0.857$.

(b) [145], Table 8, column “alle Fälle”, p. 55. If $\chi_{1}$ or $\rho_{1}$ is nonreal and $\lambda_{1} \le t$, then $\lambda_{3}>c$, for $(t,c)=(0.52,1.320)$, $(0.54,1.243)$, $(0.56,1.160)$, $(0.58,1.079)$, $(0.60,1.001)$, $(0.62,0.933)$.

(c) [145], Lemma 4.4 and Table 10, p. 55. If $\chi_{1}$ and $\rho_{1}$ are real, then $\lambda_{1} \in[0.44,0.60]$, $[0.60,0.70]$, $[0.70,0.80]$ imply $\lambda_{3}>1.176$, $1.055$, $0.952$.

(d) [145], (4.28)–(4.29), pp. 56–57. Let $\chi_{1}$ be nonreal and $\lambda_{1} \in[0.44,0.85]$, and let $f$ be the test function (11) with $\gamma=\frac{5}{4}$. Then

provided that $\sup_{y \in\mathbb{R}}\operatorname{Re}\{F(-\lambda_{1}+iy)-2F(iy)\} \le\frac{1}{6}f(0)$.

(e) [145], (4.31) and (4.34), pp. 58–59. Let $\chi_{1}$ and $\rho_{1}$ be real, $\lambda_{1} \in[0.44,0.80]$ and $\lambda_{2} \in[0.44,1.176]$, and let $f$ be (11) with $\gamma=1.04$. Then

provided that $\sup_{t \in\mathbb{R}}\operatorname{Re}\{F(-\lambda_{2}+it)-F(\lambda_{1}-\lambda_{2}+it)-F(it)\} \le\frac{5}{48}f(0)$ on this box.

In (d), the side condition is [145], (4.29), needed when a product character in his argument is principal; he verified it numerically, and we have verified it with interval arithmetic for all $\lambda_{1} \in[0.44,0.85]$: the supremum is at most $0.00335$, while $\frac{1}{6}f(0)=0.5425\ldots$. In (e), Xylouris’s argument [145], (4.32)–(4.34), p. 59 needs

on the whole box, and he states that the supremum is below 0.10, while $\frac{5}{48}f(0)=0.1351\ldots$. We have verified this with interval arithmetic as well: over $\lambda_{1}\in[0.44,0.80]$, $\lambda_{2}\in[0.44,1.18]$ and all real $t$ the supremum is at most 0.0031. (Both checks enclose the transforms on a grid of heights with step $\frac{1}{100}$ up to 30, with second-derivative interpolation errors in all variables, and use a closed-form bound beyond height 30.) (Xylouris derives eq:4.28 from Heath-Brown’s (19)–eq:10.6, whose derivation does not use the assumption $\lambda_{3}\le\frac{6}{7}$ made in [51], §10; he himself applies it to prove bounds above $\frac{6}{7}$.)

Proposition 7.5 (Parent rows). The 58 parent rows $(t,A_{j},B_{j},p_{j},r_{j})$ of Table 5, from which the roots of the case tree are built (Section 11.3), are valid implications: for $q\ge q_{0}$, every configuration of type $t$ with $\lambda_{1}\in[A_{j},B_{j}]$ has $\lambda'\ge p_{j}$ and $\lambda_{2}\ge r_{j}$.

Type	$[A_j,B_j]$	$p_j$	$r_j$	Type	$[A_j,B_j]$	$p_j$	$r_j$
rr	$[0.1,0.12]$	$2.293$	$2.56$	rc	$[0.628,0.66]$	$1.67$	$0.93$
rr	$[0.12,0.14]$	$2.293$	$2.39$	rc	$[0.66,0.7]$	$1.59$	$0.93$
rr	$[0.14,0.16]$	$2.293$	$2.25$	rc	$[0.7,0.74]$	$1.52$	$0.93$
rr	$[0.16,0.18]$	$2.293$	$2.12$	rc	$[0.74,0.78]$	$1.46$	$0.93$
rr	$[0.18,0.2]$	$2.293$	$2.01$	rc	$[0.78,0.82]$	$1.40$	$0.82$
rr	$[0.2,0.25]$	$2.293$	$1.77$	rc	$[0.82,0.86]$	$1.35$	$0.82$
rr	$[0.25,0.3]$	$2.293$	$1.58$	rc	$[0.86,0.9]$	$1.30$	$0.82$
rr	$[0.3,0.348]$	$2.195$	$1.42$	rc	$[0.9,0.94]$	$1.25$	$0.82$
rr	$[0.348,0.35]$	$2.195$	$1.42$	rc	$[0.94,0.98]$	$1.21$	$0.82$
rr	$[0.35,0.4]$	$2.108$	$1.29$	rc	$[0.98,1.02]$	$1.17$	$0.82$
rr	$[0.4,0.45]$	$2.03$	$1.18$	rc	$[1.02,1.06]$	$1.13$	$0.82$
rr	$[0.45,0.5]$	$1.958$	$1.08$	rc	$[1.06,1.099]$	$1.099$	$0.82$
rr	$[0.5,0.55]$	$1.893$	$1.0$	rc	$[1.099,1.5]$	$1.099$	$0.82$
rr	$[0.55,0.6]$	$1.832$	$0.92$	complex	$[0.44,0.58]$	$1.36$	$1.04$
rr	$[0.6,0.65]$	$1.776$	$0.85$	complex	$[0.58,0.64]$	$1.15$	$0.85$
rr	$[0.65,0.7]$	$1.724$	$0.79$	complex	$[0.64,0.66]$	$1.08$	$0.79$
rr	$[0.7,0.745]$	$1.676$	$0.745$	complex	$[0.66,0.68]$	$1.02$	$0.74$
rr	$[0.745,0.75]$	$1.676$	$0.745$	complex	$[0.68,0.70]$	$0.96$	$0.704$
rr	$[0.75,0.8]$	$1.63$	$0.745$	complex	$[0.70,0.72]$	$0.93$	$0.702$
rr	$[0.8,0.85]$	$1.587$	$0.8$	complex	$[0.72,0.74]$	$0.91$	$0.72$
rr	$[0.85,0.9]$	$1.547$	$0.8$	complex	$[0.74,0.76]$	$0.89$	$0.74$
rr	$[0.9,0.95]$	$1.509$	$0.8$	complex	$[0.76,0.78]$	$0.86$	$0.74$
rr	$[0.95,1]$	$1.473$	$0.8$	complex	$[0.78,0.80]$	$0.84$	$0.78$
rr	$[1,1.05]$	$1.439$	$0.8$	complex	$[0.80,0.82]$	$0.83$	$0.78$
rr	$[1.05,1.1]$	$1.406$	$0.8$	complex	$[0.82,1.5]$	$0.827$	$0.82$
rr	$[1.1,1.15]$	$1.375$	$0.8$
rr	$[1.15,1.175]$	$1.36$	$0.8$
rr	$[1.175,1.2]$	$1.346$	$0.8$
rr	$[1.2,1.225]$	$1.331$	$0.8$
rr	$[1.225,1.25]$	$1.318$	$0.8$
rr	$[1.25,1.275]$	$1.304$	$0.8$
rr	$[1.275,1.294]$	$1.294$	$0.8$
rr	$[1.294,1.5]$	$1.294$	$0.8$

Proof. Each entry is at most the largest of the following bounds, each valid on the whole interval $[A_{j},B_{j}]$ for the type $t$: (i) the applicable rows $(t,c)$ of Inputs 7.2 and 7.3 with $t\ge B_{j}$; (ii) the bound $c$ of an applicable row $(t,c)$ with $c=t$ (for $\lambda_{1}\le t$ the row gives it, and for $\lambda_{1}>t$ the trivial bound gives $\lambda',\lambda_{2}\ge\lambda_{1}>t=c$), as for the rows $(0.827,0.827)$ of Table $2'$ and $(1.099,1.099)$ of Table 3 of [145] and $(0.745,0.745)$ of Table 7 of [51]; (iii) the bounds of Input 7.2(b),(d) and Input 7.3(d), which hold for every $\lambda_{1}$; and (iv) the trivial bounds $\lambda',\lambda_{2}\ge\lambda_{1}\ge A_{j}$. $\square$

The positivity method

The use of nonnegative trigonometric polynomials to locate zeros goes back to de la Vallée Poussin [28]; it underlies all zero-free regions for Dirichlet $L$-functions [43, 74, 97, 82, 68] and for $\zeta$ [67, 90], and, combined with smoothed explicit formulas, the tables of Heath-Brown and Xylouris and their refinements [79]. All the new zero-location results come from the following consequence of [145], Lemmas 3.1 and 3.4. Lemma 3.4 there is a “working version” of Input 3.4: if $f$ satisfies Conditions 1 and 2, $\chi\ne\chi_{0}$ and $s\in R(9l)$, and $A_{1}$ is the set of zeros $\rho\in R(l)$ of $L(s,\chi)$ with $\operatorname{Re}\rho>\operatorname{Re}s$ (with multiplicity, at most $N$ of them) and $A_{2}$ is any set of at most $N$ zeros $\rho\in R(l)$ with $\operatorname{Re}\rho\le\operatorname{Re}s$, then for every $\varepsilon>0$ and $q\ge q_{0}(f,\varepsilon,N)$

Lemma 7.6 (Positivity method). Let $f$ satisfy Conditions 1 and 2, with Laplace transform $F$. Fix a nonnegative anchor $a$, and suppose $a\le\lambda_{1}$. Let finitely many triples $(c_{j},\chi^{(j)},t_{j})$ be given, with $c_{j}>0$, $\chi^{(j)}$ a character modulo $q$ and $|t_{j}|\le9l$, such that

For each nonprincipal $\chi^{(j)}$ let $A_{2}(j)$ be a set of at most $N$ zeros of $L(s,\chi^{(j)})$ in $R(l)$, with multiplicity. Then for every $\varepsilon>0$ and $q\ge q_{0}$,

Proof. Multiply the hypothesis by $\Lambda(n)n^{-(1-a/\mathcal{L})}f(\log n/\mathcal{L})\ge0$ and sum over $n$ (for $(n,q)>1$ every term vanishes, the principal ones included). For principal $\chi^{(j)}$ use Input 3.3 at $1-a/\mathcal{L}+it_{j}$. For the others use (12) at $s_{j}=1-a/\mathcal{L}+it_{j}$, which lies in $R(9l)$; there $A_{1}=\varnothing$, because every zero in $R(l)$ of a nonprincipal $L$-function has parameter at least $\lambda_{1}\ge a$. Divide by $\mathcal{L}$. $\square$

In every application the retained zeros satisfy $\lambda_{\rho}\ge a$, the transform $F$ is decreasing on $\mathbb{R}$, and the terms that we cannot evaluate are bounded over all real heights with interval arithmetic. A necessary inequality between the parameters of the retained zeros results, and a case is excluded when that inequality fails with a positive margin. Since there are finitely many such inequalities, each with a fixed positive margin, one $q_0$ serves all of them.

A sharper conductor coefficient

Heath-Brown remarks [51] that his estimate for $\lambda_2$ in the case of a real first character can be improved by replacing the conductor coefficient $\frac{11}{24}$ by $\frac{97}{216}$; he describes the ingredients but gives no proof. We need a weighted form of this refinement, and we prove it here. For a modulus $q$ put

We also write $\operatorname{rad}q=\prod_{p\mid q}p$.

Lemma 7.7 (Growth bounds with conductor dependence). Let $k \ge3$ and $\varepsilon> 0$. For every nonprincipal character $\chi$ modulo $q$,

where $\phi_q(\chi)=\frac{1}{4}\frac{\log(4\operatorname{rad}q)}{\log q}$ if $\chi$ is real and $\phi_q(\chi)=\min\left(\frac{1}{3},\frac{1}{4}+\frac{3}{4}\theta_q\right)$ otherwise.

Proof. Let $\chi$ be induced by the primitive character $\chi^{*}$ modulo $q^{*}$; then $q^{*}\mid q$ and $\lvert L(s,\chi)\rvert\ll q^{\varepsilon}\lvert L(s,\chi^{*})\rvert$ for $\sigma\ge\frac{1}{2}$ [51]. Heath-Brown proves ([51], (2.2)–(2.4)) that a bound $\sum_{N<n\le N+H}\chi^{*}(n)\ll_{\varepsilon,k}Q^{(k+1)/(4k^{2})+\varepsilon}q^{*\varepsilon}H^{1-1/k}$ for $1\le H\le q^{*}$ gives, by partial summation with the cut $q_1=Q^{(k+1)/(4k)}$ and the Pólya–Vinogradov inequality beyond $q^{*}$, $L(s,\chi^{*})\ll Q^{(1-\sigma)(1+k^{-1})/4}q^{*\varepsilon}(1+\lvert t\rvert)$ in the stated range (if $q_1>q^{*}$ the trivial bound up to $q^{*}$ is already stronger). For $\sigma\le1$ a bound of this form with a smaller $Q$ is stronger; for $1<\sigma\le1+\log\mathcal{L}/\mathcal{L}$ every factor $Q^{(1-\sigma)c}$ with $1\le Q\le q^{4}$ and $0\le c\le1$ lies between $\mathcal{L}^{-4}$ and $1$, so all such bounds agree up to a factor $q^{\varepsilon}$.

If $\chi$ is real, then $\chi^{*}$ is a real primitive character, so $q^{*}$ is the absolute value of a fundamental discriminant and $q^{*}\le4\operatorname{rad}(q^{*})\le4\operatorname{rad}(q)$. Its order is $2$, so by [51], which bounds the sums with the factor $l^{3/(2k)}$ for a character of order $l$, the character-sum bound holds with $Q=q^{*}$ for every $k$. This gives the real case.

In general, [51] gives the bound with $Q^{(k+1)/(4k^{2})}=v^{*(3k-1)/(4k^{2})}q^{*(k+1)/(4k^{2})}\le(v^{*3}q^{*})^{(k+1)/(4k^{2})}$, where $v^{*}$ is the cube-full part of $q^{*}$; since $q^{*}\mid q$ we have $v^{*}\mid v$ and $v^{*3}q^{*}\le v^{3}q$. This gives the exponent $\frac{1}{4}\left(1+\frac{3\log v}{\log q}\right)=\frac{1}{4}+\frac{3}{4}\theta_q$. The exponent $\frac{1}{3}$ is [51]. Taking the smaller bound gives the general case. (Bounds that are uniform in $q$ and $t$ together are studied in [48]; here the factor $1+\lvert t\rvert$ is harmless, since $\lvert t\rvert\le\mathcal{L}+1$.) $\square$

Lemma 7.8 (The refined coefficient in the explicit formula). For every $\varepsilon>0$, Input 3.4 and (12) remain true, with a suitable $q_0$, when $\phi(\chi)$ is replaced by $\phi_q(\chi)+\varepsilon$, with the radius $\delta$ that the proof of Input 3.4 provides for $\phi=\frac{1}{3}$ (which serves all characters).

Proof. Both are deduced from Heath-Brown’s Lemma 3.1 [51, 145], whose proof uses the bound (2.5) of his Lemma 2.5 only through the inequality

for $\frac{\pi}{2}\le\vartheta\le\frac{3\pi}{2}$, $R\le1/k$ and large $q$. By Lemma 7.7 this holds with $\phi=\phi_q(\chi)$, with an implied constant independent of $q$ and $\chi$. The remaining choices in that proof ($k=3+\left[3\phi/(2\varepsilon_0)\right]$, $R=1/k$, and the radius) may be made with the upper bound $\phi\le\frac{1}{3}$ in place of $\phi$, so they do not depend on $q$. The deduction of Lemma 5.2 of [51] (Input 3.4) and of (12) from Lemma 3.1 is unchanged. $\square$

Proposition 7.9 (A coupled saving in the conductor term). Let $\chi_1$ be a real nonprincipal character and let $\chi_2$ be a character with $\chi_2^{k}\ne\chi_0$ and $\chi_1\chi_2^{k}\ne\chi_0$ for $k\in K$, a finite set of positive integers. Let $c_k>0$ $(k\in K)$ with $\Sigma=\sum_k c_k\ge\frac{1}{9}$. Suppose the polynomial $1+\sum_{k\in K}c_k\cos(k\vartheta)$ is nonnegative for every real $\vartheta$. In an application of Lemma 7.6 to $(1+\chi_1(n))(1+\sum_{k\in K}c_k\operatorname{Re}((\chi_2(n)n^{-i\gamma_2})^k))$, the conductor term may, for every fixed $\varepsilon>0$ and sufficiently large $q$, be taken to be

Proof. The nonprincipal characters are $\chi_1$ with coefficient $1$, and $\chi_2^{k}$ and $\chi_1\chi_2^{k}$, each with coefficient $c_k$. By Lemma 7.8 the conductor term is at most $\frac{f(0)}{2}\left(\phi_q(\chi_1)+2\Sigma\max_{\chi}\phi_q(\chi)\right)+\varepsilon$, the maximum being over the characters $\chi_2^k$ and $\chi_1\chi_2^k$. Since $p\le(p^e)^{1/3}$ when $e\ge3$, $\operatorname{rad}q\le qv^{-2/3}$, so $\phi_q(\chi_1)\le\frac14-\frac{\theta_q}{6}+o(1)$, and $\phi_q(\chi)\le\min(\frac13,\frac14+\frac34\theta_q)+o(1)$ for every nonprincipal $\chi$ (for a real $\chi$ because $\frac14-\frac{\theta_q}{6}\le\frac14$). The conductor term is therefore at most $\mathcal{K}(\theta_q)f(0)+2\varepsilon$ for large $q$, with

On $[0,\frac19]$ the slope of $\mathcal{K}$ is $\frac34\Sigma-\frac1{12}\ge0$, and on $[\frac19,1]$ it is $-\frac1{12}$. Hence $\mathcal{K}(\theta_q)\le\mathcal{K}(\frac19)=\frac18+\frac{\Sigma}{3}-\frac1{108}$. $\square$

For $\Sigma=1$ this is Heath-Brown’s $\frac{97}{216}=\frac{11}{24}-\frac1{108}$.

Lower bounds for $\lambda_2$ when the first zero is real.

Proposition 7.10 (Second-family bounds for a real first zero). Let $\chi_1,\rho_1$ be real with $\lambda_1\in[a,b]$, where $0\le a\le b$, and let $h\ge b$. Let $P(x)=1+c_1x+c_2(2x^2-1)+c_5(16x^5-20x^3+5x)$, that is $P(\cos\vartheta)=1+c_1\cos\vartheta+c_2\cos2\vartheta+c_5\cos5\vartheta$, with

so that $P\ge0$ on $[-1,1]$ and $\Sigma=c_1+c_2+c_5\ge\frac19$. Choose $f=f_\gamma$ from (11), with transform $F$ and fixed $\gamma>0$. If $\lambda_2\le h$, then for every $\varepsilon>0$ and all sufficiently large $q$, the inequality corresponding to the character relation below must hold. The generic case means that $\chi_2$ is nonreal and none of the relations in (ii)–(iii) holds.

(i) (generic) $F(b-a)+c_1F(h-a)\le F(-a)+(\frac18+\frac{\Sigma}{3}-\frac1{108})f(0)+\varepsilon$;

(ii) $(\chi_2^2=\chi_1)$ $F(b-a)+c_1F(h-a)\le F(-a)+(\frac{1+c_2}{8}+\frac{\Sigma}{3}+\frac1{c_1+c_5})f(0)+\sup_\Delta R(\Delta)+\varepsilon$, where $R(\Delta)=c_2\{\operatorname{Re}F(-a+i\Delta)-\operatorname{Re}F(\lambda_1-a+i\Delta)\}-c_1\operatorname{Re}F(\lambda_2-a+i\Delta)$, the supremum over $\Delta\in\mathbb{R}$, $\lambda_1\in[a,b]$ and $\lambda_2\in[a,h]$;

(iii) $(\chi_2^5\in\{\chi_0,\chi_1\})$ $F(b-a)+c_1F(h-a)\le(1+c_5)F(-a)+(\frac{1+c_5}{8}+\frac{c_1+c_2}{4})f(0)+\varepsilon$;

(iv) $(\chi_2\ \mathrm{real})$ $F(b-a)+F(h-a)\le F(-a)+\frac38f(0)+\varepsilon$, with a second test function (11).

Here $\chi_2$ is a character of the second family. The test parameter in (iv) may be chosen independently of the one in (i)–(iii). If all four possible cases are excluded with a positive margin at $\varepsilon=0$, then $\lambda_2>h$ for all sufficiently large $q$.

Proof. Put $z_n=\chi_2(n)n^{-i\gamma_2}$ and apply Lemma 7.6 with anchor $a$ to $(1+\chi_1(n))P(\operatorname{Re}z_n)\ge0$ (with $\operatorname{Re}z_n^k=T_k(\operatorname{Re}z_n)$ for $|z_n|=1$), retaining $\rho_1$ in the term of $\chi_1$ and $\rho_2$ in the term $c_1\operatorname{Re}z_n$. The heights are at most $5l\le9l$. The nonzero frequencies are 1, 2 and 5, and $\chi_2\ne\chi_1$. A product character is principal only if $\chi_2$ is real, or $\chi_2^2=\chi_1$ (order 4), or $\chi_2^5\in\{\chi_0,\chi_1\}$ (order 5 or 10); the last two cannot occur together, since together they would give $\chi_2^2=\chi_0$. Generic case: there is no principal product, and Proposition 7.9 gives (i), using that $F$ is decreasing, $\lambda_1\le b$ and $\lambda_2\le h$. Case $\chi_2^2=\chi_1$: then $\chi_1\chi_2^2=\chi_0$ at height $2\gamma_2$, $\chi_2^2=\chi_1$ at height $2\gamma_2$ retains $\rho_1$, and $\chi_1\chi_2=\overline{\chi_2}$ retains $\overline{\rho_2}$; with $\Delta=2\mu_2$ these give $R(\Delta)$, while the characters of frequencies 1 and 5 are nonprincipal and are charged $\frac13$ (conservatively, since some of them have order 4). Case $\chi_2^5\in\{\chi_0,\chi_1\}$: exactly one term is principal, bounded by $c_5F(-a)$ since $f\ge0$, and every other character has order at most $10\le L$, so $\phi=\frac14$. Real $\chi_2$: apply the method to $(1+\chi_1(n))(1+\operatorname{Re}z_n)\ge0$, whose three nonprincipal characters are real. $\square$

The rows used are given in Table 6: 22 cells of width 0.0025 covering $[0.700,0.755]$. For each cell $[a,b]$ we give $h$ and the test parameter $\gamma$ of the generic case; each of the four inequalities fails by the margin shown (normalized by $f(0)$), so that $\lambda_2>h$ throughout the cell. The margins were computed with interval arithmetic: the correlation term $R(\Delta)$ is enclosed on a grid in $\Delta\in[0,30]$ with second-derivative interpolation errors in all variables, and by a closed-form bound for $|\Delta| \ge30$. The nonnegativity of $P$ was verified at the points $x=j/2000$ with a second-derivative bound, and independently by exact Bernstein expansion. In addition, on cells inside $[0.700,0.7025]$ we use the degree-2 row $P(\cos\vartheta)=(\frac{7}{8}+\cos\vartheta)^2/\frac{81}{64}$ without the conductor refinement, which gives $\lambda_2>0.762$ (Proposition 7.11).

cell $[a,b]$	$h$	$\gamma$	generic	order 4	order 5, 10	real
$[.7000,.7025]$	$.7793$	$1.09624$	$1.02\cdot10^{-4}$	$.0804$	$.1280$	$.0289$
$[.7025,.7050]$	$.7783$	$1.09522$	$5.86\cdot10^{-5}$	$.0803$	$.1279$	$.0287$
$[.7050,.7075]$	$.7772$	$1.09419$	$8.66\cdot10^{-5}$	$.0804$	$.1280$	$.0285$
$[.7075,.7100]$	$.7762$	$1.09317$	$4.92\cdot10^{-5}$	$.0803$	$.1279$	$.0284$
$[.7100,.7125]$	$.7751$	$1.09216$	$8.35\cdot10^{-5}$	$.0804$	$.1280$	$.0282$
$[.7125,.7150]$	$.7741$	$1.09115$	$5.21\cdot10^{-5}$	$.0803$	$.1279$	$.0280$
$[.7150,.7175]$	$.7730$	$1.09014$	$9.25\cdot10^{-5}$	$.0804$	$.1279$	$.0278$
$[.7175,.7200]$	$.7720$	$1.08914$	$6.71\cdot10^{-5}$	$.0804$	$.1279$	$.0276$
$[.7200,.7225]$	$.7710$	$1.08814$	$4.46\cdot10^{-5}$	$.0803$	$.1279$	$.0275$
$[.7225,.7250]$	$.7699$	$1.08715$	$9.41\cdot10^{-5}$	$.0804$	$.1279$	$.0273$
$[.7250,.7275]$	$.7689$	$1.08615$	$7.75\cdot10^{-5}$	$.0804$	$.1279$	$.0271$
$[.7275,.7300]$	$.7679$	$1.08517$	$6.39\cdot10^{-5}$	$.0804$	$.1279$	$.0269$
$[.7300,.7325]$	$.7669$	$1.08418$	$5.31\cdot10^{-5}$	$.0804$	$.1279$	$.0267$
$[.7325,.7350]$	$.7659$	$1.08321$	$4.53\cdot10^{-5}$	$.0804$	$.1279$	$.0265$
$[.7350,.7375]$	$.7649$	$1.08223$	$4.03\cdot10^{-5}$	$.0804$	$.1278$	$.0263$
$[.7375,.7400]$	$.7639$	$1.08126$	$3.83\cdot10^{-5}$	$.0804$	$.1278$	$.0261$
$[.7400,.7425]$	$.7629$	$1.08029$	$3.91\cdot10^{-5}$	$.0804$	$.1278$	$.0260$
$[.7425,.7450]$	$.7619$	$1.07933$	$4.28\cdot10^{-5}$	$.0804$	$.1278$	$.0258$
$[.7450,.7475]$	$.7609$	$1.07837$	$4.93\cdot10^{-5}$	$.0804$	$.1278$	$.0256$
$[.7475,.7500]$	$.7599$	$1.07741$	$5.87\cdot10^{-5}$	$.0804$	$.1278$	$.0254$
$[.7500,.7525]$	$.7589$	$1.07645$	$7.09\cdot10^{-5}$	$.0804$	$.1278$	$.0252$
$[.7525,.7550]$	$.7579$	$1.07550$	$8.60\cdot10^{-5}$	$.0804$	$.1278$	$.0250$

Proposition 7.11 (A second-family bound near 0.70). If $\chi_1,\rho_1$ are real and $0.700\le\lambda_1\le0.7025$, then $\lambda_2>0.762$ for $q\ge q_0$.

Proof. As in Proposition 7.10 with $c_1=\frac{112}{81}$, $c_2=\frac{32}{81}$, $c_5=0$, the printed conductor coefficients and $\gamma=1.09$ (and $\gamma=0.82$ for a real $\chi_2$); the normalized margins of the generic, order-4 and real cases are at least $3.6\cdot10^{-4}$, $0.082$ and $0.033$. $\square$

Lower bounds for $\lambda_2$ and $\lambda'$ when the first character is nonreal

For $t>0$ put

so that $c_1/c_2=4t$.

Proposition 7.12 (Second-family bounds for a nonreal first character). Let $\chi_1$ be nonreal, let $\lambda_1\in[a,b]$ with $0<a<b$, and let $h>b$. Fix $t>0$, let $c_1,c_2$ be the coefficients of $P_t$ above, and let $f=f_\gamma$ with $\gamma>0$ and transform $F$. Suppose $\lambda_2\le h$. For every $\varepsilon>0$ and sufficiently large $q$, the following necessary inequalities hold:

(i) if $\chi_2$ is nonreal and

then

(ii) if $\chi_{2}$ is real, choose $t' > 0$ and let $c'_{1}, c'_{2}$ be the coefficients of $P_{t'}$. If (13) holds with $t'$ in place of $t$ and $[a,b]$ in place of $[a,h]$, then $c'_{1}F(b-a)+F(h-a)\leq F(-a)+\left(\frac{1}{8}+\frac{c'_{1}+c'_{2}}{3}\right)f(0)+\varepsilon$.

Proof. (i) Apply Lemma 7.6 to $P_{t}(\chi_{1}(n)n^{-i\gamma_{1}})P_{t}(\chi_{2}(n)n^{-i\gamma_{2}})\geq0$, expanded as

The indices $(\pm1,0)$ retain $\rho_{1}$ (or $\overline{\rho_{1}}$) and $(0,\pm1)$ retain $\rho_{2}$ (or $\overline{\rho_{2}}$). If no nonzero index is principal, the conductor term is at most $\frac{1}{6}((1+c_{1}+c_{2})^{2}-1)f(0)$. A nonzero principal index $w$ is not on an axis, since $\chi_{1}$ and $\chi_{2}$ are nonreal, and does not have $\lvert w_{1}\rvert,\lvert w_{2}\rvert\leq1$, since $\chi_{2}\notin\{\chi_{1},\overline{\chi_{1}}\}$. So some $\lvert w_{i}\rvert=2$; choose such $i$, with $i=1$ in case of a tie, and put $T(w)=w-\operatorname{sgn}(w_{i})e_{i}$. Then $\chi^{T(w)}=\chi_{i}^{-\operatorname{sgn}w_{i}}$ is nonprincipal and not an axis index of length one, its coefficient is $4t$ times that of $w$, and the zero of $\chi_{i}^{-\operatorname{sgn}w_{i}}$ selected at the axis index sits at relative height $w\cdot\mu$. The map $T$ is injective: a collision could only come from $(2s,\sigma)$ and $(s,2\sigma)$ with $s,\sigma=\pm1$, and if both were principal then $\chi_{1}^{s}=\chi_{2}^{\sigma}$, which is excluded. Retaining at $T(w)$ the zero of $\chi_{i}^{-\operatorname{sgn}w_{i}}$, the principal term at $w$ and the extra retained term together change the inequality by at most $a_{w}\{\operatorname{Re}F(-a+iy)-\frac{1}{6}f(0)-4t\operatorname{Re}F(\lambda_{i}-a+iy)\}\leq0$, by (13). (ii) Use $P_{t'}(\chi_{1}(n)n^{-i\gamma_{1}})(1+\operatorname{Re}\chi_{2}(n)n^{-i\gamma_{2}})\geq0$. Now $\chi_{2}$ is real, with coefficient 1 and $\varphi=\frac{1}{4}$; the only possible principal indices are $(\pm2,\pm1)$, which are matched to $(\pm1,\pm1)$ with target $\overline{\rho_{1}}$ or $\rho_{1}$, as before. $\square$

Proposition 7.13 (Additional-zero bounds for a nonreal first character). Let $\chi_{1}$ be nonreal of order at least 5, let $\lambda_{1}\in[a,b]$ with $0\leq a\leq b$, and suppose $\lambda'\leq h$. Fix $t,\gamma>0$, let $c_{1},c_{2}$ be the coefficients of $P_{t}$, and let $F$ be the transform of $f=f_{\gamma}$. Choose an admissible $\rho'$ of $\chi_{1}$ with parameter $\lambda'$ and height $\gamma'$. Put $x=\lambda_{1}-a$, $y=\lambda'-a$ and $\Delta=\mathcal{L}(\gamma_{1}-\gamma')$, and

Then for every $\varepsilon>0$ and all sufficiently large $q$,

where

Proof. Apply Lemma 7.6 to $P_{t}(\chi_{1}(n)n^{-i\gamma_{1}})P_{t}(\chi_{1}(n)n^{-i\gamma'})\geq0$, whose terms are $\chi_{1}^{k+l}$ at heights $k\gamma_{1}+l\gamma'$, $\lvert k\rvert,\lvert l\rvert\leq2$. Since the order of $\chi_{1}$ is at least 5 and $\lvert k+l\rvert\leq4$, the principal terms are exactly those with $k+l=0$; they have total mass $p_{1}$ at height $\pm\Delta$ and $p_{2}$ at $\pm2\Delta$. In every term with $k+l=\pm1$ retain both $\rho_{1}$ and $\rho'$ (or their conjugates); a multiple zero is retained with its multiplicity. The terms $(1,0),(0,1)$ and their conjugates give $-c_{1}[F(x)+F(y)]-c_{1}\operatorname{Re}[F(x+i\Delta)+F(y+i\Delta)]$, and the terms $(2,-1),(-1,2)$ and their conjugates give $-\frac{1}{2}c_{1}c_{2}\operatorname{Re}[F(x+i\Delta)+F(y+i\Delta)+F(x+2i\Delta)+F(y+2i\Delta)]$. $\square$

For $\chi_{1}$ of order 3 or 4, Input 7.3(b) gives $\lambda'>1.35$ whenever $\lambda_{1}\leq0.86$, which is stronger than every bound in Proposition 7.13 that we use.

The rows used are the following, each on a cell of width 0.0025 and each with a positive normalized margin. There are 25 rows of Proposition 7.12, on cells covering $[0.66,0.7225]$, with conclusions from $\lambda_{2}>0.763$ (on $[0.66,0.6625]$) to $\lambda_{2}>0.725$ (on $[0.72,0.7225]$), parameters $\gamma\in\{1.25,1.3\}$, $t\in\{0.9,0.95\}$ and, for a real $\chi_{2}$, test parameter $\gamma\in\{1.1,1.15\}$ and $t'=0.85$; their generic margins are at least $8.2 \cdot10^{-4}$, and the margins in (13) at least 0.163. There are 69 rows of Proposition 7.13, on 69 of the 70 cells of width 0.0025 in $[0.68,0.855]$ (all except $[0.6975,0.70]$), with conclusions from $\lambda' > 0.975$ (on $[0.68,0.6825]$) to $\lambda' > 0.855$ (on $[0.8525,0.855]$), parameters $\gamma\in\{1.05,1.1,1.15\}$ and $t \in\{0.9,0.95\}$, and margins at least $1.5 \cdot10^{-3}$. For these rows, if $\lambda' \le h$, the left side of Proposition 7.13 is at least $c_1[F(b-a)+F(h-a)]$, since $F$ is decreasing, and $R(\Delta)$ is replaced by its supremum over all real $\Delta$, $x \in[0,b-a]$ and $y \in[p-a,h-a]$, where $p=\max(p_j,a)\le\lambda'$ comes from the bound $p_j$ of the parent row of the cell (Proposition 7.5; it comes from [145], Table 2′) and the trivial bound $\lambda' \ge\lambda_1 \ge a$: $p=0.96$, 0.93, 0.91, 0.89, 0.86, 0.84, 0.83 on the cells in $[0.68,0.70]$, $[0.70,0.72]$, $\ldots$, $[0.80,0.82]$, $p=0.827$ on $[0.82,0.8275]$, and $p=a$ on the cells beyond. So each row proves that $\lambda_1 \in[a,b]$ and $\lambda' \ge p$ imply $\lambda' > h$, and since $\lambda' \ge p$ holds for every configuration with $\lambda_1 \in[a,b]$, $\lambda_1 \in[a,b]$ implies $\lambda' > h$. The complete list is in Appendix A. On 11 further cells the inequalities of Proposition 7.12 are evaluated on the actual cell of the case tree.

A positivity exclusion.

Proposition 7.14 (Excluding a nearby real second character). Let $\chi_1,\rho_1$ be real with $\lambda_1 \in[a,b]$, and let $\chi_2 \ne\chi_1$ be a real nonprincipal character with a zero of height at most 1 and parameter $\nu\le h$. If

for a test function (11), then this situation does not occur for $q \ge q_0$.

Proof. Apply Lemma 7.6 to $(1+\chi_1(n))(1+\operatorname{Re}\chi_2(n)n^{-i\gamma_2})\ge0$. Its three nonprincipal characters $\chi_1$, $\chi_2$, $\chi_1\chi_2$ are real, so $\varphi=\frac{1}{4}$; retain $\rho_1$ and the given zero of $\chi_2$, which lies in $R(l)$. This gives $F(\lambda_1-a)+F(\nu-a)\le F(-a)+\frac{3}{8}f(0)+\varepsilon$. □

It excludes 197 nodes of the case tree (Section 2.4), all of type $\mathrm{rr}$ with a reserved real family; the smallest normalized margin is $7.8 \cdot10^{-5}$.

Applying the rows.

Lemma 7.15 (Preservation of configurations under location updates). In the notation of Definition 11.2, suppose that a result of this section proves $\lambda_2>h$ for every configuration of type $\mathbf{t}$ with $\lambda_1 \in[a',b']$, and let $\mathfrak{s}$ be a specification of type $\mathbf{t}$ with cell $[a,b]\subseteq[a',b']$. Then replacing $l_2$ and $r$ by $\max(l_2,h)$ and $\max(r,h)$ (and $\operatorname{lo}_2$ by the new $r$) does not change $C(\mathfrak{s})$, and $C(\mathfrak{s})=\varnothing$ if $\mathfrak{s}$ is reserved with $\operatorname{hi}_2\le h$. Similarly a proof of $\lambda'>h$ allows $p$ and $\operatorname{lo}'$ to be replaced by $\max(\mathord{\cdot},h)$, and makes $C(\mathfrak{s})$ empty if $\operatorname{hi}'\le h$.

Proof. Every family other than the first has all its zeros in $R(l)$ at parameter at least $\lambda_2>h$; in particular $\nu(\mathcal{F})>h$. □

The third family.

Proposition 7.16 (Lower bounds for the third family). Consider a configuration satisfying a specification of type $\mathbf{t}$ with $\lambda_1 \in[a,b]$. Put $h=\operatorname{hi}_2$ if a family is reserved and $h=\infty$ otherwise; then $\lambda_2\le h$ by Lemma 11.3. Each applicable rule below gives a strict lower bound for $\lambda_3$, for all sufficiently large $q$. Let $\lambda_3^{\mathrm{lo}}$ be the maximum of the finitely many bounds selected for the leaf. Then $\lambda_3>\lambda_3^{\mathrm{lo}}$.

1. 0.857 (Input 7.4(a));

2. for $\mathbf{t}=\mathrm{rr}$ and $[a,b]$ inside $[0.44,0.60]$, $[0.60,0.70]$ or $[0.70,0.80]$: 1.176, 1.055, 0.952 (Input 7.4(c));

3. for $\mathbf{t}\ne\mathrm{rr}$ and $b\le t$: the value $c$ of the smallest applicable row of Input 7.4(b);

4. for $t=\mathrm{complex}$ and $0.44 \le a \le b \le0.85$: any $t$ with

$(\gamma=\frac{5}{4})$;

5. for $t=\mathrm{rr}$, $0.44 \le a \le b \le0.80$ and $t \in[0.952,1.176]$: any $t$ such that $[a,\min(t,h)]$ is covered by intervals $[c,c']$ with

$(\gamma=\frac{26}{25})$.

Proof. (1)–(3) are printed. For (4), suppose $\lambda_{3} \le t$; then $\lambda_{2} \le\min(t,h)$. The function $\lambda\mapsto F(-\lambda)-F(t-\lambda)=\int f(u)e^{\lambda u}(1-e^{-tu})\,\mathrm{d}u$ is increasing, so $F(-\lambda_{1})-F(\lambda_{3}-\lambda_{1}) \le F(-\lambda_{1})-F(t-\lambda_{1}) \le F(-b)-F(t-b)$, and $F(\lambda_{2}-\lambda_{1}) \ge F(\min(t,h)-a)$. So the right side of Input 7.4(d) is below $-10^{-5}f(0)+\varepsilon<0$, a contradiction; the side condition [145] (4.29) holds by our verification. For (5), suppose $\lambda_{3} \le t \le1.176$. Then $\lambda_{2} \in[a,\min(t,h)] \subseteq[0.44,1.176]$ and $\lambda_{1} \in[0.44,0.80]$, so Input 7.4(e) applies. If $\lambda_{2} \in[c,c']$, the same monotonicity gives $F(-\lambda_{2})-F(\lambda_{3}-\lambda_{2}) \le F(-c')-F(t-c')$ and $F(\lambda_{1}-\lambda_{2}) \ge F(b-c)$, and the right side of (e) is negative for small $\varepsilon$, a contradiction; the side condition [145] (4.34) holds by our verification. $\square$

For example, on the $\mathrm{rr}$ cell $[0.7025,0.705]$ rule (5) gives $\lambda_{3}>1.0502$ without a reservation, and $\lambda_{3}>1.0888$ with a reserved family below $0.85$; Table 10 gives $0.952$. Rules (4) and (5) are evaluated with interval arithmetic, and a value $t$ is accepted only with a normalized margin exceeding $10^{-5}$.

The graded near-density lemma

The purpose of this section is to constrain many characters at once while retaining the location of each selected zero. We first bound a quadratic form in smoothed prime sums, called responses (Theorem 8.4). We then bound each response from below using selected zeros (Lemma 8.7). Finally, an elementary optimization argument introduces the single threshold used in a near row (Proposition 8.8). The analytic inputs are stated in Section 3; the scope of the formalization is described in Section 14.3. Section 8.3 explains the idea of the proof, and Example 8.9 shows what the resulting constraint can do that separate counts cannot.

Data

The detector determines the response to a zero. The Gram test and sieve weight control correlations between characters. The anchors may vary between entries, but all test functions and allowed offsets are fixed before $q$ tends to infinity.

Definition 8.1 (Near-density data). A near datum consists of the following fixed objects; Conditions 1 and 2 are those of Definition 3.1.

• A detector $f$ satisfying Condition 1, with transform $F$.

• A Gram test $g$ satisfying Conditions 1 and 2, with transform $G$.

• Real numbers $s_{1}$ and $s$, called the safe anchor and the reference anchor.

• A sieve weight: cells $t_{0}<t_{1}<\cdots<t_{m}$ with $t_{0}>\frac{1}{3}+2\varepsilon'$ for a fixed $\varepsilon'>0$, and heights $h_{0},\ldots,h_{m-1}\ge0$. We put

and require that $\omega$ be bounded below by a positive constant on the support of $f$.

• A finite set $\Delta\subset[0,\infty)$ of offsets.

We put $\vartheta_{k}=\frac{1}{2}(t_{k}-\frac{1}{3})-\varepsilon'$ (the sieve levels), so $0<\vartheta_{k}<\frac{1}{2}t_{k}$, and

Here the integrand of $I_B$ is taken to be 0 where $f=0$. We require $I_B>0$.

The number $I_B$ is the first Cauchy–Schwarz factor, $D(\delta)$ bounds a diagonal entry at offset $\delta$, and $d_1$ bounds the correlation between distinct characters. Heath-Brown’s Lemma 12.1 is the formal analogue of the case $g=f$, $s=s_1=\lambda_1$, $\Xi=0$ and all offsets 0 (in that case $\omega=fe^{s_1t}$ is not bounded below on the support of $f$, and Heath-Brown writes $f=\sqrt{f}\cdot\sqrt{f}$ instead).

Definition 8.2 (Entries and responses). An entry is a triple $(\chi_j,\gamma_j,\delta_j)$ of a nonprincipal character modulo $q$, a real height $\gamma_j$ and an offset $\delta_j\in\Delta$. Its response anchor is $s_j=s-\delta_j$, and its response is

For a family of entries, the pair excess of entry $j$ is

Definition 8.3 (Safe anchor). For $\Gamma'>0$, the safe-anchor hypothesis $\operatorname{SA}(s_1,\Gamma')$ says that no nonprincipal $L(s,\chi)$ modulo $q$ has a zero $\rho$ with $|\operatorname{Im}\rho|\leq\Gamma'$ and $\operatorname{Re}\rho>1-s_1/\mathcal{L}$.

The pair excess $e_j$ is needed only when two entries use the same character. It records the part of their correlation that exceeds the bound $d_1$ for distinct characters. The safe-anchor hypothesis ensures that discarded zero terms in the Gram estimate have the required sign.

In the application $s_1\leq\lambda_1$. Then $\operatorname{SA}(s_1,2l+1)$ holds for large $q$, by the definition of $\lambda_1$ and Input 3.5: a zero with $|\gamma|\leq2l+1\leq10l$ and $\operatorname{Re}\rho>1-s_1/\mathcal{L}$ lies in $R(10l)$, hence in $R(l)$, so its parameter is at least $\lambda_1\geq s_1$.

Statement

Theorem 8.4 (Graded near-density lemma). Assume Inputs 3.3, 3.4, 3.7 and 3.8, Mertens’ estimate, and the divisor bound (the number of divisors of $q$ is $O_\varepsilon(q^\varepsilon)$). Fix a near datum as in Definition 8.1 and a tolerance $\eta'>0$. There is a threshold $q_0$, depending only on this fixed data, with the following property.

For $q\geq q_0$, let $J$ be a finite index set and let $(\chi_j,\gamma_j,\delta_j)_{j\in J}$ be entries satisfying

1. $|\gamma_j|\leq\Gamma$ for some $0\leq\Gamma\leq\mathcal{L}/3$;

2. $\operatorname{SA}(s_1,2\Gamma+1)$;

3. the characters $\chi_j$ are pairwise distinct whenever $\Xi\ne0$.

Then, for every choice of real coefficients $a_j\geq0$,

The responses $R_j$ and pair excesses $e_j$ are those of Definition 8.2. The threshold $q_0$ is uniform in the number of entries, their characters and heights, and the coefficients $a_j$.

Two special cases of the pair excess are used.

1. Separated heights. If $\chi_k=\chi_j$ implies $|\gamma_j-\gamma_k|\mathcal{L}\geq M$ for $k\ne j$, and $M$ is so large that $\operatorname{Re}G(-\sigma+iY)\leq d_1$ for $|Y|\geq M$ and $\sigma$ in the finite set of values $\sigma_{jk}$, then $e_j=0$ (Lemma 8.6; this holds once $g(0)>0$).

2. A controlled pair. If entry $j$ shares its character with exactly one other entry $k$, with $\delta_j=\delta_k=0$ and $|(\gamma_j-\gamma_k)\mathcal{L}|$ in a range on which $\operatorname{Re}G(-s_1+iy)\leq c^{\mathrm{hi}}$, then $e_j\leq(c^{\mathrm{hi}}-d_1)_{+}$.

The idea of the proof

Each response $R_j$ is a smoothed sum over primes, twisted by $\chi_j(n)n^{-i\gamma_j}$ and evaluated at the point $1-s_j/\mathcal{L}+i\gamma_j$. By the explicit formula (Lemma 8.7 below), every zero $\rho$ of $L(s,\chi_j)$ near $1+i\gamma_j$ contributes about $\operatorname{Re} F((\lambda_\rho-s_j)+i(\gamma_j\mathcal{L}-\mu_\rho))$ to $R_j$, and this is large when $\lambda_\rho$ is small. So characters with zeros close to 1 have large responses.

On the other hand, all the responses are correlations of one vector, the weighted primes, with the twisted vectors $(\chi_j(n)n^{-i\gamma_j})_n$. By the Cauchy–Schwarz inequality a combination $\sum_j a_jR_j$ is at most the length of the prime vector, which gives the factor $I_B$, times the length of $\sum_j a_j\chi_j(n)n^{-i\gamma_j}$ measured with the weight $\omega$. The latter is a Gram form in the characters. Its diagonal entries are the numbers $D(\delta_j)$, and distinct characters are nearly orthogonal on the primes: their correlation is at most $d_1=g(0)/6$. Indeed, by the explicit formula the correlation is governed by the zeros of the quotient character $\chi_j\overline{\chi_k}$ to the right of the safe anchor, and the safe-anchor hypothesis says that there are none. Hence only a bounded number of characters can have large responses.

Heath-Brown’s Lemma 12.1 is the case in which all entries are tested at the same anchor. The grading gives each entry its own anchor $s_j=s-\delta_j$. An entry tested at a larger offset $\delta_j$ has a smaller diagonal $D(\delta_j)$, because of the factor $e^{-2\delta_jt}$, so characters whose zeros are further from 1 are charged less. The sieve weight $\Xi$ adds Selberg-sieve majorants of the primes to $\omega$ on the ranges $[q^{t_k},q^{t_{k+1}})$. This makes $\omega$ larger and the first factor $I_B$ smaller. The price is a sieve contribution to the diagonal, computed with Graham’s asymptotic, and off-diagonal sums of the quotient characters over long intervals. These are small by Burgess’s estimate provided the quotient characters are nonprincipal, that is, provided the characters are distinct.

A Burgess bound for all nonprincipal characters

Lemma 8.5 (Burgess bound for imprimitive characters and long intervals). Assume Input 3.7. For every $\varepsilon>0$ there is $C$ such that for every nonprincipal character $\chi$ modulo $q$ and all real $N\geq0$ and $H\geq1$,

Proof. Let $\chi$ be induced by the primitive character $\chi^*$ modulo $q^*>1$, and let $r$ be the product of the primes dividing $q$ but not $q^*$. Then $\chi(n)=\chi^*(n)\sum_{e\mid(n,r)}\mu(e)$, so the sum is $\sum_{e\mid r}\mu(e)\chi^*(e)\sum_{N/e<m\leq(N+H)/e}\chi^*(m)$. The sum only depends on the integers in the range, so we may take the endpoints real. In each inner sum remove complete periods of $\chi^*$, over which $\chi^*$ sums to 0 because $\chi^*$ is nonprincipal; what remains is a sum over at most $\min(H/e,q^*)$ consecutive integers, which by periodicity may be shifted to start at a point $\geq1$. Input 3.7 bounds it by $Cq^{*1/9+\varepsilon}\min(H/e,q^*)^{2/3}\leq Cq^{1/9+\varepsilon}\min(H,q)^{2/3}$ (for an interval of length less than 1 the bound is trivial). By the divisor bound, the number of $e$ is $O_\varepsilon(q^\varepsilon)$. $\square$

Proof of Theorem 8.4

Write $t_n=\log n/\mathcal{L}$. For $(n,q)=1$ put

Since $n^{-(1-s_j/\mathcal{L}+i\gamma_j)}=n^{-1}e^{st_n}e^{-\delta_jt_n}n^{-i\gamma_j}$, and $\chi_j(n)=0$ for $(n,q)>1$,

Cauchy–Schwarz. Since $\omega>0$ on the support of $f$ and $\omega\ge0$ everywhere,

The first factor. The function $\varphi_B(t)=e^{2s_1t}f(t)^2/\omega(t)$ is bounded, because $\omega$ is bounded below on the support of $f$; it vanishes outside the support of $f$ and is continuous except at finitely many points, so it is Riemann integrable. Given $\varepsilon_B>0$ choose a partition $0=u_0<\cdots<u_K$ of an interval containing the support of $f$ whose upper Riemann sum is at most $I_B+\varepsilon_B/2$. By Mertens’ estimate $\sum_{n\le x}\Lambda(n)/n=\log x+O(1)$ (see [112] for explicit forms) we have $\mathcal{L}^{-1}\sum_{q^{u_i}<n\le q^{u_{i+1}}}\Lambda(n)/n=u_{i+1}-u_i+O(1/\mathcal{L})$, so the first factor is at most $I_B+\varepsilon_B$ for large $q$.

The second factor. Split it as $Q_1+Q_2$ according to $\omega=ge^{s_1t}+\Xi$. Expanding $|Y_n|^2$ and writing $\chi_{jk}=\chi_j\overline{\chi_k}$,

All the points $w_{jk}=1-\sigma_{jk}/\mathcal{L}+i(\gamma_j-\gamma_k)$ involved satisfy $|\operatorname{Re}w_{jk}-1|\le C/\mathcal{L}$ and $|\operatorname{Im}w_{jk}|\le2\Gamma\le\mathcal{L}$, so Inputs 3.3 and 3.4 apply uniformly. The error terms $o(1)$ below are uniform in the characters, heights and entries.

• Diagonal, $j=k$. Here $\chi_{jj}=\chi_0$ and the height is 0. By Input 3.3,

  $$X_{jj}=\int_0^\infty g(t)e^{(s_1-2\delta_j)t}\,\mathrm{d}t+o(1).$$

• Distinct characters. Here $\chi_{jk}$ is nonprincipal. By Input 3.4,

  $$X_{jk}\le-\sum_\rho\operatorname{Re}G((w_{jk}-\rho)\mathcal{L})+\frac{\varphi(\chi_{jk})}{2}g(0)+o(1),$$

  the sum running over the zeros of $L(s,\chi_{jk})$ in a disc of radius $\delta<1$ about $1+i(\gamma_j-\gamma_k)$. Such a zero has $|\operatorname{Im}\rho|\le2\Gamma+1$, so by $\mathrm{SA}(s_1,2\Gamma+1)$ it has $\operatorname{Re}\rho\le1-s_1/\mathcal{L}\le1-\sigma_{jk}/\mathcal{L}=\operatorname{Re}w_{jk}$. Hence $\operatorname{Re}(w_{jk}-\rho)\mathcal{L}\ge0$, and each term $\operatorname{Re}G(\cdot)$ is $\ge0$ by Condition 2. Dropping them and using $\varphi\le\frac13$, $g(0)\ge0$, we get $X_{jk}\le d_1+o(1)$. When $\sigma_{jk}<0$ the point $w_{jk}$ lies to the right of 1, and the same argument applies.

• The same character, $j\ne k$. Here $\chi_{jk}=\chi_0$ at the height $\gamma_j-\gamma_k$. By Input 3.3,

  $$X_{jk}=\operatorname{Re}G(-\sigma_{jk}+i(\gamma_j-\gamma_k)\mathcal{L})+o(1).$$

For $a\ge0$ the off-diagonal part is therefore at most

Since $\epsilon_{jk}=\epsilon_{kj}$ ($\operatorname{Re}G$ is even in the imaginary part, $g$ being real) and $2a_ja_k\le a_j^2+a_k^2$, we have $\sum_{j\ne k}a_ja_k\epsilon_{jk}\le\sum_je_ja_j^2$. Also $\sum_{j\ne k}a_ja_k=(\sum_ja_j)^2-\sum_ja_j^2$. Hence

The sieve part $Q_2$. It is present only when $\Xi\ne0$, so the characters are distinct and all pair excesses vanish. Fix a cell $k$, let $z_k=q^{\vartheta_k}$ and let

• Majorant. If $p$ is prime with $t_p \ge t_k$ then $p \ge q^{t_k} > z_k$, so the only divisor of $p$ up to $z_k$ is 1 and $\nu_k(p)=\xi_1^2=1$. Hence $\Lambda(p)=(\log p)\nu_k(p)$. For $n$ not a prime power, $\Lambda(n)=0\le(\log n)\nu_k(n)$. Prime powers $p^e$ with $e\ge2$ and $t_{p^e}\ge t_k$ contribute $\ll\mathcal{L}q^{-t_k/2}(\sum_j a_j)^2=o(1)(\sum_j a_j)^2$. So the contribution of cell $k$ to $Q_2$ is at most

where now $|Y_n|^2$ is expanded without the coprimality condition, with $\chi_j(n)=0$ for $(n,q)>1$.

• Diagonal. The terms $j=k'$ of $|Y_n|^2$ are at most $a_j^2e^{-2\delta_jt_n}$ with $\chi_j\overline{\chi_j}\le1$. By Input 3.8, $\sum_{n\le x}\nu_k(n)=x/\log z_k+O(x/\log^2z_k)$ for $x\ge z_k$. Partial summation gives

• Off-diagonal. For $j\ne k'$ the character $\chi_{jk'}=\chi_j\overline{\chi_{k'}}$ is nonprincipal, since the characters are distinct. Put $W(n)=(\log n)n^{-1}e^{-(\delta_j+\delta_{k'})t_n}n^{-i(\gamma_j-\gamma_{k'})}$. Opening $\nu_k$, the term is

where $m$ runs over an interval $[M_0,M_1)$ with $M_0=q^{t_k}/[d,e]\ge q^{t_k-2\vartheta_k}=q^{1/3+2\varepsilon'}$. By Lemma 8.5, $|\sum_{M_0\le m\le u}\chi_{jk'}(m)|\ll u^{2/3}q^{1/9+\varepsilon}$ for every real $u\ge M_0$. We also have $|W(u[d,e])|\ll\mathcal{L}(u[d,e])^{-1}$ and $|\partial_uW(u[d,e])|\ll\mathcal{L}(1+|\gamma_j-\gamma_{k'}|)u^{-2}[d,e]^{-1}\ll\mathcal{L}^2u^{-2}[d,e]^{-1}$. Abel summation, with the bound for the partial sums $\sum_{M_0\le m\le u}\chi_{jk'}(m)$ at every $u$, therefore bounds the inner sum by

Since $\sum_{d,e\le z_k}[d,e]^{-2/3}\le\sum_{g\le z_k}g^{-2/3}(\sum_{d'\le z_k/g}d'^{-2/3})^2\le9\zeta(\frac{4}{3})z_k^{2/3}$ and $|\xi_d|\le1$, the term is

because $2\vartheta_k=t_k-\frac{1}{3}-2\varepsilon'$. With $\varepsilon<\varepsilon'/2$ this is $o(1)$ uniformly, and the sum over $j,k'$ is $o(1)(\sum_j a_j)^2$. (A bound for the partial sums only at the end point of the interval, combined with the total variation of $W$, would lose a factor $(M_1/M_0)^{2/3}$, which is too large for the cells used.)

Summing over the cells,

Conclusion. By (15) and (16) the second factor is at most $B+o(1)(\sum_j a_j)^2$, where $B=\sum_j(D(\delta_j)-d_1+e_j)a_j^2+d_1(\sum_j a_j)^2\ge\sum_j(D(\delta_j)+e_j)a_j^2\ge0$. With the first factor $I_B+o(1)$ and $I_B>0$, the right side of (14) is at most $(1+\eta')I_B[B+\eta'(\sum_j a_j)^2]$ for $q$ large. $\square$

Responses and the threshold form.

Lemma 8.6 (Uniform decay in the imaginary direction). If $f$ satisfies Condition 1, then $\operatorname{Re}F(x+iY)\to0$ as $|Y|\to\infty$, uniformly for $x$ in compact sets.

Proof. Integrating by parts twice, $F(z)=f(0)/z+O_K(|z|^{-2})$ for $\operatorname{Re} z$ in a compact set $K$. Moreover $\operatorname{Re}(f(0)/z)=f(0)\operatorname{Re} z/|z|^2=O_K(|z|^{-2})$. $\square$

Lemma 8.7 (Response lemma). Assume Input 3.4. Let $f$ satisfy Conditions 1 and 2, and fix $\sigma_{\max}>0$, an integer $K_0\geq0$, and $\eta'>0$. There are $M>0$, $\delta\in(0,1)$ and $q_0$ such that the following holds for $q\geq q_0$, and it remains true when $\delta$ is replaced by any smaller radius (with a suitable $q_0$). Let $\chi$ be nonprincipal, $|\sigma|\leq\sigma_{\max}$ and $|\gamma|\leq\mathcal{L}$. Let $S$ be a finite set of distinct zeros of $L(s,\chi)$ in the disc $|1+i\gamma-\rho|\leq\delta$, each retained with its full multiplicity $m_\rho$. Suppose that every other zero of $L(s,\chi)$ in the disc with $\lambda_\rho<\sigma$ satisfies $|\gamma\mathcal{L}-\mu_\rho|\geq M$, and that there are at most $K_0$ such zeros, counted with multiplicity. Then the response of $(\chi,\gamma,\sigma)$, that is $R=-\mathcal{L}^{-1}\sum_n\Lambda(n)\operatorname{Re}(\chi(n)n^{-(1-\sigma/\mathcal{L}+i\gamma)})f(t_n)$, satisfies

Proof. By Lemma 8.6 choose $M$ with $|\operatorname{Re}F(x+iY)|\leq\eta'/(2K_0+2)$ for $|Y|\geq M$ and $x\in[-\sigma_{\max},0]$. Apply Input 3.4 with $\varepsilon=\eta'/2$ at the point $w=1-\sigma/\mathcal{L}+i\gamma$, and let $\delta$ be its disc radius, or any smaller radius (see the remark after Input 3.4). We have $(w-\rho)\mathcal{L}=(\lambda_\rho-\sigma)+i(\gamma\mathcal{L}-\mu_\rho)$, so

the sum counting multiplicity. Zeros in $S$ are kept. A zero outside $S$ with $\lambda_\rho\geq\sigma$ has $\operatorname{Re}(w-\rho)\mathcal{L}\geq0$ and a term $\geq0$ (Condition 2), which we drop. The remaining zeros have $\lambda_\rho\in[0,\sigma)$ and $|\gamma\mathcal{L}-\mu_\rho|\geq M$; there are at most $K_0$ of them, and each term is at least $-\eta'/(2K_0+2)$. Finally $\varphi(\chi)\leq\frac{1}{3}$ and $f(0)\geq0$. $\square$

Proposition 8.8 (Threshold form). Let $J$ be a finite index set, let $D_j>0$ and $v_j\in\mathbb{R}$ for $j\in J$, and let $d>0$. Suppose that

Then there is $\tau\in[0,\sqrt d]$ with

We call the $v_j$ the features, the $D_j$ the diagonals and $d$ the correlation term of the inequality, and $\tau$ its threshold. A feature measures how strongly an entry is detected, a diagonal what the entry costs on its own, and $d$ bounds the correlation between distinct entries.

Example 8.9 (One level and two levels). (i) Suppose that $N$ entries have the same feature $v>0$ and the same diagonal $D$, with $v^2>d$. Taking every $a_j=1$ in the hypothesis of Proposition 8.8 gives $(Nv)^2\leq ND+dN^2$, that is,

This is the kind of bound that Heath-Brown’s Lemma 12.1 provides: a bound for the number $N(\lambda)$ of characters with a zero at distance at most $\lambda$, all counted alike.

(ii) Now take $D=1$ and $d=\frac{1}{4}$, one entry with feature $v_1=1$ (a zero close to 1) and two entries with feature $v_2=\frac{3}{4}$ (zeros further away). The count of (i), applied to each level separately, allows this: it permits up to $D/(v_1^2-d)=\frac{4}{3}$ entries with feature at least $v_1$, and up to $D/(v_2^2-d)=\frac{16}{5}$ with feature at least $v_2$. But (T) fails for every $\tau$: for $0 \le\tau\le\frac{1}{2}$ its left side minus its right side is

whose discriminant $25-\frac{63}{2}$ is negative. (Equivalently, taking all three $a_j=1$ violates the hypothesis: $\left(1+\frac{3}{2}\right)^2=\frac{25}{4}>3+\frac{9}{4}$.) So this configuration is excluded. A single threshold couples the near entry and the farther ones, which separate counts cannot do. In the leaf programs this coupling acts on all the bins of characters at once.

Proof. The function $\mathcal{J}(\tau)=\tau^2/d+\sum_j(v_j-\tau)_+^2/D_j$ is convex and $C^1$ on $[0,\infty)$, and tends to $\infty$; let $\tau^*$ minimize it. If $\tau^*=0$ then $\mathcal{J}'(0)\ge0$, so all $v_j\le0$ and $\mathcal{J}(0)=0$. Otherwise $\mathcal{J}'(\tau^*)=0$, that is $\tau^*=d\sum_j a_j$ with $a_j=(v_j-\tau^*)_+/D_j\ge0$. Then $\sum_j a_jv_j=\sum_j a_j(v_j-\tau^*)+\tau^*\sum_j a_j=\sum_jD_ja_j^2+d(\sum_j a_j)^2=\mathcal{J}(\tau^*)$. The hypothesis gives $\mathcal{J}(\tau^*)^2\le\mathcal{J}(\tau^*)$, so $\mathcal{J}(\tau^*)\le1$. Finally (T) forces $\tau^2\le d$. $\square$

Corollary 8.10 (Zero form). Fix a near datum whose detector $f$ also satisfies Condition 2. Fix a common bound $K_0$ for the omitted zeros in Lemma 8.7, and choose $\sigma_{\max}\ge\max_{\delta\in\Delta}|s-\delta|$ with $\sigma_{\max}>0$. Choose $\eta'>0$, $I_u\ge I_B$, and $D_u>0$, and put

For sufficiently large $q$, consider entries satisfying Theorem 8.4 with tolerance $\eta''$. For each entry $j$, suppose the retained set $S_j$ satisfies Lemma 8.7 at anchor $s_j$, with the separation and disc radius chosen for tolerance $\eta''$. Define

and choose $D_j^+>0$ with $D_j^+\ge D(\delta_j)-d_1+e_j$ for every $j$. Then there is $\tau\in[0,\sqrt d]$ satisfying (T) with

The same threshold remains valid if features $v_j$ are decreased, diagonals $D_j$ or the correlation term $d$ are increased, or nonpositive features are replaced by $0$. An entry with nonpositive feature may also be omitted.

Proof. Apply Theorem 8.4 and Lemma 8.7 with the tolerance $\eta''$ fixed in the statement. The response lemma gives $R_j\ge k_j\coloneqq r_j-\eta''$, so for $a\ge0$ we have $(\sum_j a_jk_j)_+\le(\sum_j a_jR_j)_+$. Dividing the inequality of Theorem 8.4 by $I_BD_u$, we get for all $a\ge0$

where we also replaced each coefficient $D(\delta_j)-d_1+e_j$ by the larger $D_j^+$. Proposition 8.8 gives $\tau$ satisfying (T) with these features, diagonals and correlation term; only the positivity of the $D_j^+$ is needed, not that of $D(\delta_j)-d_1+e_j$. We compare with the displayed quantities.

• If $r_j>0$ then $v_j\le k_j/\sqrt{I_BD_u}$, because $I_u\ge I_B$ and $\eta''/\sqrt{I_BD_u}\le\eta'$. If $r_j\le0$ then $v_j<0$.

• $(1+\eta')\ge(1+\eta'')$, so the displayed diagonals are at least the ones above.

• $(1+\eta')(d_1+\eta'D_u)\ge(1+\eta'')(d_1+\eta'')$, since $\eta''\le\eta'D_u$.

Finally, if (T) holds at $\tau$ for features $v_j$, diagonals $D_j$ and correlation term $d$, it also holds at $\tau$ for features $v'_j \le\max(v_j,0)$ with $v'_j \le v_j$ or $v'_j \le0$, diagonals $D'_j \ge D_j$ and correlation term $d' \ge d$. Indeed each term $(v'_j-\tau)_+^2/D'_j$ is at most $(v_j-\tau)_+^2/D_j$ (the former is 0 when $v'_j \le0 \le\tau$), $1-\tau^2/d \le1-\tau^2/d'$, and $\tau\le\sqrt d \le\sqrt{d'}$. $\square$

The near rows

A near row is a constraint obtained by applying the threshold inequality (T) to selected characters and zeros. In a leaf program (Section 2.4), characters are grouped by parameter into bins, each with its own column. Known entries from the first or reserved family may instead be recorded separately as family terms. There are four kinds of rows, the family, shifted, graded and two-test rows; Table 8 in Section 9.5 lists their anchors, sieve weights and entries.

We first specify the test functions and normalization. We then justify each permitted kind of entry and describe how to combine them into rows. Most rows follow from Corollary 8.10; the two-test row in Section 9.6 has a separate proof. Proposition 9.7 is the conclusion needed later: every actual zero configuration in a leaf satisfies each of its near rows at some threshold.

The parabolic test function

All detectors and Gram tests are normalized parabolic autocorrelations. For $c>0$ put

Thus $\widehat f_{2\gamma}=f_\gamma/f_\gamma(0)$ with $f_\gamma$ as in (7.1). Write $\widehat F_c$ for its transform.

Lemma 9.1 (Properties of the parabolic test functions). Let $c>0$.

(i) $\widehat f_c\ge0$, $\widehat f_c(0)=1$, $\widehat f_c$ is decreasing on $[0,c]$, and $\widehat f_c$ extended by $0$ is twice continuously differentiable on $[0,\infty)$ with $|\widehat f_c''|\le10/c^2$. So Condition 1 holds.

(ii) $\operatorname{Re}\widehat F_c(iy)=60c(cy\cos\frac{cy}{2}-2\sin\frac{cy}{2})^2/(cy)^6\ge0$ for $y\ne0$, and Condition 2 holds.

(iii) $x\mapsto\widehat F_c(x)$ is positive and strictly decreasing on $\mathbb R$.

(iv) For $z\ne0$, $\widehat F_c(z)=c\Phi(cz)$, where

The singularity at $0$ is removable. Consequently, for $d\ge0$ and $|y|\ge U>0$,

where $R(u,E)=\frac{10}{u^3}+\frac{30}{u^4}+\frac{120}{u^6}+E\left(\frac{30}{u^4}+\frac{120}{u^5}+\frac{120}{u^6}\right)$.

(v) For $d\ge0$ put $C_c(d)=\sup_{\operatorname{Re}z\ge-d}(-\operatorname{Re}\widehat F_c(z))$. It is finite,

and $C_c(0)=0$.

Proof. (i) is elementary: $P'(u)=-5u(2+u)(1-u)^2$, and $P(1)=P'(1)=P''(1)=0$. (ii) The function $\widehat f_c$ is a positive multiple of the autocorrelation of $x\mapsto(\gamma^2-x^2)_+$ with $c=2\gamma$, so its transform on the imaginary axis is a positive multiple of $\left(\int_0^\infty(\gamma^2-x^2)_+\cos(xy)\,\mathrm dx\right)^2$ [145], Lemma 3.6, which gives the closed form. Condition 2 then follows from Input 3.2 with $F_2=0$, since $\widehat F_c(z)\to0$ uniformly in $\operatorname{Re}z>0$. (iii) $\widehat F_c'(x)=-\int t\widehat f_c(t)e^{-xt}\,\mathrm dt<0$. (iv) Five integrations by parts, using the values of $P$ and its derivatives at 0 and 1. For $z=-d+iy$ we have $c\operatorname{Re}\frac{1}{cz}=\frac{-d}{d^{2}+y^{2}}$, which is $\le0$ and $\ge-d/U^{2}$, and each other term is bounded by the corresponding term of $cR(c|z|,e^{cd})$, which decreases in $|z|\ge U$. (v) $-\operatorname{Re}\widehat{F}_{c}(z-d)$ is harmonic in $\operatorname{Re}z>0$, continuous up to the boundary and tends to 0 uniformly at infinity, so its supremum over $\operatorname{Re}z\ge0$ is attained on the boundary or is 0; $C_{c}(0)=0$ by Condition 2. $\square$

Row data

Every family, shifted and graded row fixes rational numbers $\gamma_{f}>0$ (detector), $\gamma_{g}>0$ (Gram test), $s_{1}\le s$ (safe and reference anchors), $t_{0}$ and $\mu_{\mathrm{row}}$, and uses the near datum of Section 8.1 with $f=\widehat{f}_{2\gamma_{f}}$, $g=\widehat{f}_{2\gamma_{g}}$, $\varepsilon'=\frac{1}{2000}$, and 2000 equal cells $t_{0}<t_{1}<\cdots<t_{2000}=2\gamma_{f}$, where $t_{0}>\frac{1}{3}+2\varepsilon'$. The heights $h_{k}\ge0$ are rational numbers. They are chosen by a heuristic that brings $\omega$ close to the Cauchy–Schwarz optimum $\omega\propto e^{st}f$: the height of cell $k$ is the positive part of $e^{sm_{k}}f(m_{k})/\sqrt{\mu_{\mathrm{row}}\kappa_{k}}-g(m_{k})e^{s_{1}m_{k}}$ at the midpoint $m_{k}$, where $\kappa_{k}=(t_{k+1}^{2}-t_{k}^{2})/(2\vartheta_{k}(t_{k+1}-t_{k}))$ is the mean sieve cost per unit length of the cell, computed in binary64 arithmetic and converted to a rational number with denominator at most $10^{12}$. Any nonnegative heights are admissible, so this choice plays no role in the proofs. In every family and shifted row $\mu_{\mathrm{row}}=10^{9}$ and all the computed heights are 0, so $\Xi\equiv0$. The offsets are the finitely many numbers $s-\sigma$ that occur for the anchors $\sigma$ of the entries. In every row the lower bound of $\omega$ on each cell of the support of $f$ is checked to be positive; in the rows with $\Xi\equiv0$ this holds because $\gamma_{g}>\gamma_{f}$.

We use the normalization of Corollary 8.10: $I_{u}$ is an upper Riemann sum for $I_{B}$ over 4000 cells, computed with outward interval arithmetic; $D_{u}\ge D(0)$; each entry $j$ receives the feature $v_{j}=r_{j}/\sqrt{I_{u}D_{u}}-\eta'$ (rounded down, and replaced by 0 if negative) and the diagonal $(1+\eta')D_{j}^{+}/D_{u}$ (rounded up), where $D_{j}^{+}$ is an upper bound for $D(\delta_{j})-d_{1}+e_{j}$; and the row has the correlation term $d=(1+\eta')(d_{1}/D_{u}+\eta')$ (rounded up). Here $\eta'=10^{-6}$. For an entry with offset $\delta$ the value $D(\lfloor\delta\rfloor_{1/200})\ge D(\delta)$ is used (here $\lfloor\delta\rfloor_{1/200}$ is $\delta$ rounded down to a multiple of $\frac{1}{200}$, and $D$ is decreasing), and an entry is omitted if its diagonal numerator does not meet the positivity margin used in the construction. An omitted entry has feature 0 in the row and contributes nothing to (T). The rounding directions are justified by the last assertion of Corollary 8.10.

Facts available in a leaf

A leaf specifies intervals and lower bounds for the first few zeros. The full definition of a specification $\mathfrak{s}$ and its configuration set $C(\mathfrak{s})$ is in Definition 11.2; the information needed for this section is listed below. Write $\nu(\mathcal{F})$ for the least parameter in a family among zeros of physical height at most 1, and $\nu_{T^{*}}(\chi)$ for the corresponding minimum at height at most $T^{*}$. A reserved family minimizes $\nu$ outside the first family. It need not be the family attaining the global parameter $\lambda_{2}$.

Fix a leaf with first-zero cell $[a,b]$ and a configuration in $C(\mathfrak{s})$. For all sufficiently large $q$, the following facts are available.

(Z1) $\lambda_{1}\in[a,b]$. The safe anchor of the leaf is $s_{1}=\lfloor50a\rfloor/50\le\lambda_{1}$.

(Z2) $\lambda'\ge p^{*}$, where $p^{*}=\mathrm{lo}'$ if there is a gap (an interval $[\mathrm{lo}',\mathrm{hi}']$ known to contain $\lambda'$) and $p^{*}=p$ otherwise; if there is a finite gap, $\lambda'\le\mathrm{hi}'$.

(Z3) Every zero in $R(l)$ of a character outside the first family has parameter at least $\lambda_{2}\ge l_{2}$.

(Z4) Every family $\mathcal{F}\ne\mathcal{F}_{1}$ has $\nu(\mathcal{F})\ge r$, and ordinary characters have $\nu_{T^{*}}\ge\nu\ge r$.

(Z5) If $\mathfrak{s}$ is reserved, the reserved family has $n_{2}$ characters with common height-one parameter $\nu_{*}\in[\mathrm{lo}_{2},\mathrm{hi}_{2}]$.

(Z6) Inside: $|\mu_{1}|\le C_{B}$. Outside: $|\mu_{1}|>C_{B}$.

Lemma 9.2 (No zero to the right). Let $\chi$ be nonprincipal, let $|\gamma|\le2l$ and $0<\delta\le1$, and fix a real anchor $\sigma$. For sufficiently large $q$, every zero $\rho$ of $L(s,\chi)$ in the disc $|1+i\gamma-\rho|\le\delta$ has $\lambda_{\rho}\ge\sigma$ in either of the following cases:

(a) $\sigma<\lambda_{1}$;

(b) $\chi\notin\mathcal{F}_{1}$ and $\sigma\le l_{2}$, where $l_{2}\le\lambda_{2}$ is the leaf’s global second-family lower bound.

Equivalently, this disc contains no zero to the right of the line $\operatorname{Re}s=1-\sigma/\mathcal{L}$.

Proof. Such a zero has $|\operatorname{Im}\rho|\le2l+\delta<10l$. By Input 3.5 it lies in $R(l)$ or has $\lambda_{\rho}>\frac{1}{3}\log\log\mathcal{L}$, which exceeds every fixed $\sigma$. In $R(l)$ its parameter is at least $\lambda_{1}$, and at least $\lambda_{2}\ge l_{2}$ if $\chi\notin\mathcal{F}_{1}$ (Z3). $\square$

Lemma 9.3 (Separation of zeros beyond the representative strip). Let $\chi$ be an ordinary character with $T^{*}$-representative $\rho_{\chi}$, of height $\gamma_{\chi}$ and parameter $\lambda_{\chi}$, and let $\sigma\le\min(\lambda_{\chi},s_{\max})$. Then every zero $\rho$ of $L(s,\chi)$ with $|1+i\gamma_{\chi}-\rho|\le\delta_{0}$ and $\lambda_{\rho}<\sigma$ satisfies $|\gamma_{\chi}\mathcal{L}-\mu_{\rho}|\ge M$, and there are at most $K_{0}$ such zeros, counted with multiplicity.

Proof. Such a zero has $|\operatorname{Im}\rho|\le T^{*}+\delta_{0}<2\le10l$ and $\lambda_{\rho}<\sigma\le s_{\max}<\frac{1}{3}\log\log\mathcal{L}$, so it lies in $R(10l)$ and hence, by Input 3.5, in $R(l)$. As $\lambda_{\rho}<\sigma\le\lambda_{\chi}$, the minimality of the representative gives $|\operatorname{Im}\rho|>T^{*}$; so the zero is counted in $K_{0}$. By Lemma 6.3, $|\operatorname{Im}\rho|\ge T^{*}+M/\mathcal{L}$, so $\mathcal{L}|\operatorname{Im}\rho-\gamma_{\chi}|\ge\mathcal{L}(|\operatorname{Im}\rho|-T^{*})\ge M$. $\square$

Entries

We now list the entries that occur in the rows, with their kept zeros. In each case the hypotheses of Lemma 8.7 hold with the stated kept set $S$ and anchor $\sigma$; we write $r$ for the resulting lower bound for the response, before normalization, so that the entry’s feature is $r/\sqrt{I_{u}D_{u}}-\eta'$. Table 7 summarizes the entries. Recall that $F(x)$ is positive and decreasing on $\mathbb{R}$, and that a kept zero with $\lambda_{\rho}\ge\sigma$ has $m_{\rho}\operatorname{Re}F(\mathord{\cdot})\ge\operatorname{Re}F(\mathord{\cdot})\ge0$.

(a) Ordinary character in the bin $[\mathrm{lo},\mathrm{hi})$. Entry $(\chi,\gamma_{\chi},s-\sigma)$ at its $T^{*}$-representative, with $\sigma=\min(\mathrm{lo},s)$; $S=\{\rho_{\chi}\}$. If $\sigma\le l_{2}$ there is no zero to the right (Lemma 9.2); otherwise Lemma 9.3 gives the hypotheses. Then $r=F(\mathrm{hi}-\sigma)-\frac{1}{6}f(0)$, and the diagonal numerator is $D(s-\sigma)-d_{1}$.

(b) Inside first family, one entry per character. Entries $(\chi_{1},\gamma_{1},s-\sigma)$ and, for type complex, $(\overline{\chi}_{1},-\gamma_{1},s-\sigma)$, with $\sigma=\min(a,s)\le\lambda_{1}$; $S=\{\rho_{1}\}$ (resp. $\{\overline{\rho}_{1}\}$). No zero to the right. $r=F(b-\sigma)-\frac{1}{6}f(0)$, and the diagonal numerator is $D(s-\sigma)-d_{1}$.

(c) Conjugate pair, type rc (rows with $s=s_{1}$ and $\Xi=0$). Entries $(\chi_{1},\pm\gamma_{1},0)$, a pair of entries of the same character with height difference $2\gamma_{1}$; $S=\{\rho_{1},\overline{\rho}_{1}\}$ for both. No zero to the right, and

over $x\in[a-s_{1},b-s_{1}]$ and $\mu$ in the height range of the leaf ($[0,1]$ or $[1,\infty)$). Pair excess $e\le(c^{\mathrm{hi}}-d_{1})_{+}$ with $c^{\mathrm{hi}}\ge\sup\operatorname{Re}G(-s_{1}+2i\mu)$ over the range, so the diagonal numerator is $D(0)-d_{1}+(c^{\mathrm{hi}}-d_{1})_{+}$.

(d) Second zero of the first family (rows with $s=s_{1}$ and $\Xi=0$; type complex with a finite gap $[p^{*},\mathrm{hi}']$). Let $\rho'$ be an admissible zero of $\chi_{1}$ realizing $\lambda'$ (Section 2), of height $\gamma'$, namely the one that witnesses the piece of the second-zero split (Section 11.6) in which the configuration lies, and $y=(\gamma_{1}-\gamma')\mathcal{L}$, restricted to that piece: $|y|\in[0,1]$, $[1,\frac{3}{2}]$, $[\frac{3}{2},2]$, $[2,3]$, $[3,5]$ or $[5,\infty)$. Entries $(\chi_{1},\gamma_{1},0)$, $(\chi_{1},\gamma',0)$, $(\overline{\chi}_{1},-\gamma_{1},0)$, $(\overline{\chi}_{1},-\gamma',0)$, with kept sets $\{\rho_{1},\rho'\}$, $\{\overline{\rho}',\overline{\rho}_{1}\}$ and their conjugates (a zero is kept only if it lies in the disc of the entry). No zero to the right. Then

where $R_{p}\le\inf\operatorname{Re}F(x+iy)$ over $x\in[p^{*}-s_{1},\mathrm{hi}'-s_{1}]$ and $R_{q}\le\inf\operatorname{Re}F(x+iy)$ over $x\in[a-s_{1},b-s_{1}]$, with $|y|$ in the piece. Here $x\ge0$ (as $s_{1}\le a\le p^{*}$), so $\operatorname{Re}F(x+iy)\ge0$ by Condition 2 and we take $R_{p},R_{q}\ge0$; the multiplicities of the kept zeros can then only increase the sums. On the last piece $R_{p}=R_{q}=0$, which also covers a zero outside the disc (its term is then absent). Each same-character pair is a controlled pair with excess $(c^{\mathrm{hi}}-d_{1})_{+}$, $c^{\mathrm{hi}}\ge\sup\operatorname{Re}G(-s_{1}+iy)$, so every diagonal numerator is $D(0)-d_{1}+(c^{\mathrm{hi}}-d_{1})_{+}$; an entry of $\chi_{1}$ and an entry of $\overline{\chi}_{1}$ form a pair of distinct characters, with the nonprincipal quotient $\chi_{1}^{2}$, and carry no mutual excess. If $\rho'=\rho_{1}$ is a double zero, the two entries coincide and $r\ge m_{\rho_{1}}F(\lambda_{1}-s_{1})-\frac{1}{6}f(0)$ with $m_{\rho_{1}}\ge2$, which is at least both $r_{\gamma_{1}}$ and $r_{\gamma'}$ (here $y=0$ and $\lambda'=\lambda_{1}$).

(e) Shifted first family (the shifted row, $s=\min(1.9,p^{*},l_{2})$, used only if $s>s_{1}$ and $s\ge a$). Entries $(\chi_{1},\gamma_{1},0)$ and, for type complex, $(\overline{\chi}_{1},-\gamma_{1},0)$, with anchor $s$. The only zeros of $\chi_{1}$ with parameter below $s$ are $\rho_{1}$ and, for type rc, $\overline{\rho}_{1}$, since every other occurrence has parameter at least $\lambda'\ge p^{*}\ge s$. With $S=\{\rho_{1}\}$ (and $\overline{\rho}_{1}$ for type rc) nothing remains in the hypotheses of Lemma 8.7. As conjugate zeros of a real character have the same multiplicity $m$,

with $C_{Z}=C_{c}(s-a)$ from Lemma 9.1(v) ($0$ if $s\le a$). If $F(b-s)-\mathbf{1}_{\mathrm{rc}}C_{Z}\ge0$ this is at least $F(b-s)-\mathbf{1}_{\mathrm{rc}}C_{Z}$, and we take $r=F(b-s)-\mathbf{1}_{\mathrm{rc}}C_{Z}-\frac{1}{6}f(0)$. Otherwise the proposed bound $F(b-s)-C_{Z}-\frac{1}{6}f(0)$ is negative and the entry receives the feature $0$; this is admissible whatever the true response bound $r_{j}$ is, by the last clause of Corollary 8.10 (with $v'_{j}=0$). The diagonal numerator is $D(0)-d_{1}$.

(f) Reserved family. Entries $(\chi,\gamma_{\chi},s-\sigma_{2})$ for its $n_{2}$ characters at their height-one representatives, with $\sigma_{2}=\min(\max(a,\min(l_{2},\mathrm{lo}_{2})),s)$. Every zero of these characters in $R(l)$ has parameter at least $\max(\lambda_{1},\lambda_{2})\geq\max(a,l_{2})\geq\sigma_{2}$, so there is no zero to the right, $r=F(\mathrm{hi}_{2}-\sigma_{2})-\frac{1}{6}f(0)$, and the diagonal numerator is $D(s-\sigma_{2})-d_{1}$. With second-family columns, the column $[\mathrm{lo}_{2,j},\mathrm{hi}_{2,j}]$ uses $\sigma=\min(\max(a,\min(l_{2},\mathrm{lo}_{2,j})),s)$, $r=F(\mathrm{hi}_{2,j}-\sigma)-\frac{1}{6}f(0)$ and the diagonal numerator $D(s-\sigma)-d_{1}$.

(g) Outside first family (family row, $s=s_{1}$, $\Xi=0$). Global entries $(\chi_{1},\gamma_{1},0)$ with $\mathcal{S}=\{\rho_{1}\}$ and, for type complex, $(\bar{\chi}_{1},-\gamma_{1},0)$ with $\mathcal{S}=\{\bar{\rho}_{1}\}$ (for type rc there is the single global entry $(\chi_{1},\gamma_{1},0)$); $r=F(b-s_{1})-\frac{1}{6}f(0)$. And a hidden entry $(\chi,\gamma_{h},0)$ for each first-family character $\chi$ with a zero in $R_{P}$, at its zero $\rho_{h}$ of least parameter $t_{\chi}\in[\mathrm{lo}_{h},\mathrm{hi}_{h})$, with $\mathcal{S}=\{\rho_{h}\}$ and $r=F(\mathrm{hi}_{h}-s_{1})-\frac{1}{6}f(0)$. No zero to the right. Entries of the same character have normalized height differences at least $C_{B}-C_{P}>M$, so their pair excess vanishes, and every diagonal numerator is $D(0)-d_{1}$.

The rows

Each leaf program has one or more rows, of the kinds listed in Table 8. In the family row of an inside leaf the first family is entered by (b), by (c) for type rc (with the height split of Section 11.6), or by (d) in a piece of the second-zero split. In the shifted row every ordinary anchor is at most $s\leq l_{2}$. In the graded rows all characters are distinct.

In every family, shifted and graded row the safe-anchor hypothesis holds by (Z1) and Input 3.5, as explained after Definition 8.3, and the characters are pairwise distinct whenever $\Xi\not\equiv0$. All the entries have heights at most $l$. The first family is entered once per character in a row with sieve weights (for type rc only $\rho_{1}$ is kept); the pair and second-zero entries occur only in rows with $\Xi=0$.

The two-test row

On 37 inside roots the graded rows alone do not certify, and the leaf program contains in addition the following near row. It is not an instance of Theorem 8.4: its Gram form is at the shift $s$ of the leaf, which may exceed $\lambda_{1}$, and it pays explicit corrections for the quotient characters $\chi_{1},\bar{\chi}_{1}$. Its proof follows Xylouris’s proof of his Lemma 5.3 [145], (5.34)–(5.40), pp. 72–76, with two test functions.

Fix $\gamma_{G}>\gamma_{Z}>0$ and let $f=f_{\gamma_{Z}}$ and $g=f_{\gamma_{G}}$ as in (11), so that $g>0$ on the support of $f$. Let $s\leq\min(p^{*},l_{2})$ be the shift of the leaf, and put

The function $f^{2}/g$, extended by $0$, satisfies Condition 1, since it vanishes to order $6$ at $2\gamma_{Z}$. Define the features

with $\phi_{1}=\frac{1}{4}$ for a real $\chi_{1}$ and $\phi_{1}=\frac{1}{3}$ otherwise, $\phi_{2}=\frac{1}{4}$ if $n_{2}=1$ and $\phi_{2}=\frac{1}{3}$ otherwise, and $\mathbf{1}_{\mathrm{rc}}$ the indicator of type rc; the correlation term $d=k_{\eta}(d_{0}+\eta)$; and the diagonals

caption:

:::

Theorem 9.4 (Two-test inequality). Use the two-test data defined above, with $\gamma_{G}>\gamma_{Z}>0$ and $s\leq\min(p^{*},l_{2})$. For every sufficiently large $q$ and every configuration of an inside leaf, form the following entries:

1. any finite set of distinct ordinary characters, each at a zero $\rho_{\chi}\in R(l)$ with $|\operatorname{Im}\rho_{\chi}|\leq1$;

2. the first character at $\rho_{1}$, and also $\overline{\chi}_{1}$ at $\overline{\rho}_{1}$ when $\chi_{1}$ is nonreal;

3. the $n_{2}$ characters of the reserved family at their height-one representatives, if a family is reserved.

For type rc, the single first-family entry retains both $\rho_{1}$ and $\overline{\rho}_{1}$. Assign features $v(\lambda_{\rho_{\chi}})$, $v_{f}$, and $v_{2}$ to groups (i)–(iii), and diagonals $D_{o}$, $D_{f}$, and $D_{o}$, respectively. Then for every choice of real coefficients $a_{j}\geq0$,

Proof. Put $\sigma_{s}=1-s/\mathcal{L}$ and, in the Hilbert space of sequences indexed by $n$,

Then $-\operatorname{Re}\langle A_{j},B\rangle=\mathcal{L}R_{j}$, where $R_{j}$ is the response of the entry at the anchor $s$, and $\mathcal{L}\sum_{j}a_{j}R_{j}\leq\left\|\sum_{j}a_{j}A_{j}\right\|\|B\|$.

Responses. We use Lemma 8.7 (or directly Input 3.4). For an ordinary or reserved character every zero in the disc has parameter at least $s$ (Lemma 9.2(b), as $s\leq l_{2}$), so $R_{j}\geq F(\lambda_{\rho_{\chi}}-s)-\frac{\phi}{2}f(0)-\varepsilon$. For the first family, the zeros of $\chi_{1}$ with parameter below $s$ are $\rho_{1}$ and, for type rc, $\overline{\rho}_{1}$ (the others have parameter at least $\lambda'\geq p^{*}\geq s$); keeping them gives $F(\lambda_{1}-s)\geq F(b-s)$ and, for type rc, $\operatorname{Re}F(\lambda_{1}-s+2i\mu_{1})\geq-C_{F}$ (Lemma 9.1(v)). Conjugate zeros of a real character have the same multiplicity $m\geq1$, so the kept terms total at least $m(F(b-s)-\mathbf{1}_{\mathrm{rc}}C_{F})$, which is at least $F(b-s)-\mathbf{1}_{\mathrm{rc}}C_{F}$ when this is nonnegative (otherwise $v_{f}<0$, see below). Since $\varphi(\chi)=\frac{1}{4}$ for real $\chi$ and $\varphi(\chi)\leq\frac{1}{3}$ always (Input 3.4), the terms $\frac{\varphi}{2}f(0)$ are at most $\frac{1}{6}f(0)$, $\frac{\phi_{1}}{2}f(0)$ and $\frac{\phi_{2}}{2}f(0)$ respectively. Thus $R_{j}\geq\mathcal{N}(v_{j}+\eta)-\varepsilon$ for entries with positive feature. Entries with nonpositive feature may be omitted in estimating the positive part on the left; we justify restoring their coefficients to the right-hand side below.

Norms. By Input 3.3 applied to $f^{2}/g$, $\|B\|^{2}=\mathcal{L}R_{Z}(1+o(1))$, and $\|A_{j}\|^{2}=\mathcal{L}R_{G}(1+o(1))$.

Off-diagonal terms. For $j\neq k$ the characters are distinct and $\langle A_{j},A_{k}\rangle$ is the $g$-weighted sum for the nonprincipal character $\chi_{jk}=\chi_{j}\overline{\chi}_{k}$ at $\sigma_{s}+i(\gamma_{j}-\gamma_{k})$. By Input 3.4 its real part is at most $\mathcal{L}(d_{0}R_{G}+\varepsilon)$ plus $\mathcal{L}$ times the sum of $-\operatorname{Re}G(\cdot)$ over the zeros of $\chi_{jk}$ in the disc with parameter below $s$. If $\chi_{jk}\notin\{\chi_{1},\overline{\chi}_{1}\}$ there are none, by Lemma 9.2(b), which applies because $|\gamma_j-\gamma_k|\le2\le2l$. If $\chi_{jk}=\chi_1$ (or $\overline{\chi}_1$), they are $\rho_1$ (resp. $\overline{\rho}_1$), and for type rc both, each contributing at most $C_G$; these zeros are simple, since a multiple $\rho_1$ would give $\lambda_1=\lambda'\ge s$. For a given $j$, the $k$ with $\chi_j\overline{\chi}_k\in\{\chi_1,\overline{\chi}_1\}$ are $\chi_k\in\{\chi_j\overline{\chi}_1,\chi_j\chi_1\}$: at most two, and at most one if $\chi_1$ is real; a first-family character has none if $\chi_1$ is real ($\chi_1\overline{\chi}_1=\chi_0$) and at most one ($\chi_1^2$) otherwise. With $2a_ja_k\le a_j^2+a_k^2$ for each such pair, these corrections give the diagonals of the table.

Assembly. For $a\ge0$, $\|\sum_j a_jA_j\|^2\le LR_G[\sum_j a_j^2(1-d_0+\operatorname{exc}_j)+(d_0+\varepsilon)(\sum_j a_j)^2]$, and $\|B\|^2\le(1+\varepsilon)LR_Z$. Dividing by $L^2\mathcal{N}^2$ and absorbing the errors in $\eta$ gives the claim for the positive-feature entries. To restore any omitted entries, expand the right-hand side as $\sum_j(D_j+d)a_j^2+2d\sum_{j<k}a_ja_k$. Its coefficients are nonnegative: the table gives $D_j+d\ge k_\eta(1+\eta)>0$ and $d>0$. Restoring nonnegative $a_j$ therefore increases this bound, while a nonpositive feature can only decrease the positive part on the left. □

Proposition 9.5 (Detector mixture). Theorem 9.4 remains true when $f$ is replaced by $f_\epsilon=f+\epsilon g$ with $\epsilon\ge0$, provided $F$, $f(0)$ and $C_F$ are replaced by $F+\epsilon G$, $f(0)+\epsilon g(0)$ and $C_F+\epsilon C_G$, and $\mathcal{N}$ by $\mathcal{N}_\epsilon=\sqrt{R_G(R_Z+2\epsilon F(-s)+\epsilon^2R_G)}$; the Gram quantities are unchanged.

Proof. $f_\epsilon$ satisfies Conditions 1 and 2, and $f_\epsilon^2/g=f^2/g+2\epsilon f+\epsilon^2g$ is a sum of functions satisfying Condition 1, so $\|B_\epsilon\|^2=L(R_Z+2\epsilon F(-s)+\epsilon^2R_G)(1+o(1))$. The rest of the proof is unchanged, using $-\operatorname{Re}(F+\epsilon G)\le C_F+\epsilon C_G$ on $\operatorname{Re}z\ge a-s$. □

Proposition 9.6 (Paired first family). Use the data and inside-leaf hypotheses of Theorem 9.4. Suppose $\chi_1$ is nonreal and the specification $\mathfrak{s}$ has a finite gap $\lambda'\in[p^*,h]$, where $h=h_{i'}$. Fix $\zeta>0$ and a number $\mathcal{B}$ with

for all real $\Delta$, $\lambda_1\in[a,b]$ and $\lambda'\in[p^*,h]$. Let $e\ge C_F/(2\mathcal{N})$, and put

Let $m'\le m$ and $D'\ge D_*$ with $D'>0$ and $2D'\ge\zeta_*\max(m',0)$. Then, for every configuration of the leaf, there is $\tau\in[0,\sqrt d]$ satisfying (T) for the entries of Theorem 9.4, with the first-family feature $m'$ and diagonal $D'$ in place of $v_f$ and $D_f$.

Proof. For $\chi_1$ use the averaged vector $A_f=\frac12(A_{\chi_1,\gamma_1}+A_{\chi_1,\gamma'})$, with an admissible $\rho'$ of height $\gamma'$ (Section 2; so $\rho'$ is a zero of $\chi_1$ and $|\gamma'|\le l$), and its conjugate for $\overline{\chi}_1$. Its response is the average of the two responses, which we bound by (12) at the points $1-s/\mathcal{L}+i\gamma_1$ and $1-s/\mathcal{L}+i\gamma'$; they lie in $R(9l)$, and (12) does not require the kept zeros to lie in a disc. The set $A_1$ of zeros of $\chi_1$ in $R(l)$ with parameter below $s$ is contained in $\{\rho_1\}$, since every occurrence in $R(l)$ other than the distinguished one has parameter at least $\lambda'\ge p^*\ge s$. We put $\rho'$ into $A_2$, and also $\rho_1$ if $\lambda_1\ge s$, with their multiplicities (if $\rho'=\rho_1$ this is $\rho_1$ with multiplicity at least 2); by Input 3.9 their number is bounded independently of $q$. The multiplicities can only increase the response: the terms of $\rho'$ are $\ge0$ by Condition 2 because $\lambda'\ge s$, the term $F(\lambda_1-s)$ is positive, and if $\rho_1$ is a multiple zero then $\lambda_1=\lambda'\ge s$, so its terms are $\ge0$ as well. With $\Delta=\mathcal{L}(\gamma_1-\gamma')$, $P=\operatorname{Re}G(-s+i\Delta)/R_G$ and $Q$ the second fraction in (17), the feature of $A_f$ is therefore at least $\frac{F(b-s)+F(h-s)}{2\mathcal{N}}-\frac{f(0)}{6\mathcal{N}}+Q-\eta$, and $\|A_f\|^2=LR_G\frac{1+P}{2}(1+o(1))$ by Input 3.3. Each off-diagonal term $\langle A_f,A_k\rangle$ is the average of two terms of the kind estimated in the proof of Theorem 9.4, at the heights $\gamma_1-\gamma_k$ and $\gamma'-\gamma_k$, whose absolute values are at most $l+1\le2l$; the term between $A_f$ and its conjugate is an average of four terms, at heights $2\gamma_1$, $\gamma_1+\gamma'$ and $2\gamma'$, which are also at most $2l$ in absolute value. So Lemma 9.2 applies to all of them, averaging creates no new quotient characters, and the off-diagonal estimates are unchanged (for a cubic $\chi_1$ the quotient $\chi_{1}^{2}=\overline{\chi_{1}}$ of the conjugate pair is paid by the $c_{G}$ in $D_{*}$). Put $U=Q+e$; then $U\geq0$, since $\lambda'\geq s$ gives $Q\geq-C_{F}/(2N)$.

By (17), $P\leq B+\zeta Q$, so the actual first-family feature is at least $m+U$ and the actual diagonal $k_{\eta}\left\{\frac{1+P}{2}-d_{0}+c_{G}\right\}$ is at most $D_{*}+\zeta_{*}U$. Put $U'=U+(m-m')\geq U\geq0$. Then the actual feature is at least $m'+U'$, and the actual diagonal is at most $D'+\zeta_{*}U'$, which is positive. Replacing the first-family features and diagonals by these bounds preserves the quadratic inequality of Theorem 9.4, and Proposition 8.8 gives $\tau\in[0,\sqrt{d}]$ satisfying (T) with the first-family data $(m'+U',D'+\zeta_{*}U')$. Finally, for $\tau\geq0$ and $U'\geq0$,

Indeed the right side is 0 if $m'\leq\tau$. Otherwise put $z=m'-\tau\in(0,m']$. The derivative of $(z+u)^{2}/(D'+\zeta_{*}u)$ in $u\geq0$ is $(z+u)(2D'-\zeta_{*}z+\zeta_{*}u)/(D'+\zeta_{*}u)^{2}\geq0$, because $2D'\geq\zeta_{*}m'\geq\zeta_{*}z$. So (T) holds at the same $\tau$ with $(m',D')$. $\square$

In the leaf program the two-test row is the threshold form (T) of Theorem 9.4 (or of one of its two variants, Propositions 9.5 and 9.6):

with $V_{i}=v(\mathrm{hi}_{i})$ the feature of the right end of the ordinary bin $i$ (the tail and the reserved columns have feature 0 in this row), and negative features replaced by 0. The $T^{*}$-representatives are admissible entries, since Theorem 9.4 allows any zero of height at most 1, and all of them have parameter at least $l_{2}\geq s$ (Z3). The reserved family appears as a fixed term with the feature $v_{2}$ at $\mathrm{hi}_{2}$, which bounds its feature over the whole reserved range, and with the diagonal $D_{o}$. The row is used on 37 roots, once on each: with a single test on 31 of them, with a mixture on 2, and in the paired form on 4 (all with a nonreal $\chi_{1}$, as Proposition 9.6 requires). In the paired form the certified pair $(m',D')$ consists of a lower bound for $m$ and an upper bound for $D_{*}$, and the condition $2D'\geq\zeta_{*}\max(m',0)$ is checked for this pair; the margins $2D'-\zeta_{*}m'$ are between 0.18 and 0.30. The condition (17) is verified with interval arithmetic. Both of its terms are even in $\Delta$, so $\Delta\geq0$ suffices. On the grid $\Delta\in\frac{1}{200}\mathbb{Z}\cap[0,30]$ the left side is enclosed at the four corners of the parameter box $[a,b]\times[p^{*},h]$. As the left side is a sum of a function of $\lambda_{1}$ and a function of $\lambda'$, its maximum over the box exceeds the maximum over the corners by at most $\frac{\zeta}{2N}\left(\frac{(b-a)^{2}}{8}\int t^{2}f(t)e^{(s-a)+t}\,\mathrm{d}t+\frac{(h-p^{*})^{2}}{8}\int t^{2}f(t)\,\mathrm{d}t\right)$, the linear interpolation error in each variable, and this is added. Between grid points a second-derivative bound in $\Delta$ is used, and for $\Delta\geq30$ a closed-form bound from Lemma 9.1(iv). The value $B$ used is at least 0.

Validity of the rows.

Proposition 9.7 (Every configuration satisfies the near rows). Let $s$ specify a leaf or one of its subcases. For sufficiently large $q$, take any configuration in $C(s)$ and assign its characters to columns as in Proposition 11.6, obtaining column values $x$. Then, for every near row $k$, there is a threshold $\tau_{k}$ such that

Here $d_{k}$ is the stored correlation term, $S=10^{16}$, and $\Phi_{k}$ is the row expression of Definition 10.1. Each row may have its own threshold.

Proof. For a graded, family or shifted row, take as entries the characters of the configuration that are placed in the columns and family terms of the row: the ordinary characters of $O$ at their $T^*$-representatives, the reserved characters, and the first family, with the entries of Section 9.4. These form an admissible family of entries for Theorem 8.4, with the kept sets verified in Section 9.4, so Corollary 8.10 gives a threshold $\tau$ at which $(T)$ holds with the normalized features, diagonals and correlation term. Each column’s feature is at most the feature of every character placed in it (features are taken at the right end of the column, and $F$ is decreasing), or is $0$; each column’s diagonal is at least theirs, or the column has feature $0$ (an entry whose diagonal numerator is not safely positive is omitted, and its column is given feature $0$), in which case it contributes nothing for $\tau\ge0$; each family term $(n,v,D)$ receives $n$ entries with feature at least $v$ and diagonal at most $D$; and the characters in the tail column and in the hidden tail are simply omitted from the row. Since each term $(v-\tau)_{+}^{2}/D$ increases with $v$ and decreases with $D$ for $\tau\ge0$, the row of the leaf holds at $\tau$. For the two-test row the same argument applies to Theorem 9.4 and its variants, through Proposition 8.8. $\square$

The numbers $I_u$, $D_u$, $D(\delta)$, the transforms and the special terms $R_{\mathrm{lo}}$, $c^{\mathrm{hi}}$, $R_p$, $R_q$, $C_Z$ and $C_F$ are computed from the rational parameters of each row with outward interval arithmetic. The transforms are enclosed with the closed form of Lemma 9.1(iv) or with its power series near $0$. The infima and suprema over heights in $R_{\mathrm{lo}}$, $c^{\mathrm{hi}}$, $R_p$ and $R_q$ are enclosed on grids of step at most $\frac{1}{50}$, with second-derivative bounds $\int t^2\widehat{f}_c(t)e^{dt}\,\mathrm{d}t$ between grid points and Lipschitz bounds in the real parts, and by the tail bounds of Lemma 9.1(iv) beyond the grid. The constants $C_Z$, $C_F$ and $C_G$ are enclosed on the line $\operatorname{Re} z=-d$ (by Lemma 9.1(v)), for heights in $[0,20]$ by adaptive linear interpolation, starting from $200$ cells of width $\frac{1}{10}$ and bisecting cells until the largest cell bound is within a small fixed tolerance of the best value found, with the same second-derivative bound, and beyond height $20$ by a closed-form tail bound. The features, diagonals and correlation terms of all rows other than the $37$ two-test rows have been recomputed from their rational parameters by an independent program whose soundness is formally verified (Section 14).

Leaf programs and exact certificates

For each case of the zero analysis, we construct a family of linear programs indexed by the thresholds in the near rows. Their common objective bounds the zero sum. We cover the threshold domain by finitely many boxes and give a linear relaxation on each box. An exact certificate then bounds every feasible point, without having to find an optimal one.

The argument uses weak linear-programming duality [25, 115] with outward rounding of the data; compare [63, 95, 1]. This section is independent of the analytic number theory. Its conclusion is the certificate soundness theorem, Theorem 10.7, which is formally verified (Section 14.3).

Leaf data

The coefficients, budgets, and objective constants of a leaf are stored as integers at scale

Thus a stored coefficient $v$ represents $v/S$. Character counts such as $n$, $n_g$, and $n_E$ are ordinary unscaled integers. The variables $x_c$ and thresholds $\tau_k$ are also unscaled real numbers.

A leaf has a finite set of columns $c$ and a finite list of near rows $k=1,\ldots,K$. A column $c$ carries

• an objective coefficient $G_c$ and a far weight $W_c$;

• a count coefficient $C_c$, a hidden-count coefficient $N_c$ and a second-family indicator $E_c$ (stored at scale $S$: $C_c,E_c\in\{0,S\}$, and $N_c$ is $0$, $S$ or, for the hidden tail, $\lfloor S/w(\mathrm{end})\rfloor$, with end as in Section 11.5);

• for each row $k$, a feature $v_{ck}$ and a diagonal $D_{ck}>0$.

A row $k$ carries a correlation term $d_k>0$ and finitely many family terms $(n,v,D)$, where $n\ge0$ is a count, $v$ a feature and $D>0$ a diagonal. The leaf also has a far budget $F$, counts $n_g$ (the number $n$ of first-family characters, in outside leaves) and $n_E$ (the number of reserved characters carried by second-family columns: the size of the reserved family if it is charged through columns, and 0 otherwise), and two constants: $\mathrm{first}$, which bounds the cost of the first family of an inside leaf (0 for an outside leaf) plus any fixed charge for the reserved family, and $\mathrm{final}$, the error allowance, with $\mathrm{final}/S=5\eta$ (Section 11.5).

Definition 10.1 (Leaf program). For real column values $x=(x_c)$ and real thresholds $\tau=(\tau_k)$, put

The pair $(x,\tau)$ is feasible if all of the following hold:

• $x_c\ge0$ for every $c$;

• (far) $\sum_c W_cx_c\le F$;

• (count) $\sum_c C_cx_c\le2S$;

• (hidden count) $\sum_c N_cx_c\le n_gS$;

• (second family) $\sum_c E_cx_c=n_ES$;

• (near rows) for every $k$: $0\le\tau_k$, $\tau_k^2\le d_k/S$ and $\Phi_k(x,\tau_k)\le1-\tau_k^2/(d_k/S)$.

The value of $x$ is

For fixed $\tau$, these are linear constraints on $x$. Allowing $\tau$ to vary gives the family of programs to be certified. The variables $x_c$ are real and nonnegative: integer character counts are feasible values, but the relaxation also permits fractional counts and the weighted tail masses defined in Section 11.5.

In Section 11 we show that for every configuration of zeros the actual characters give a feasible pair whose value bounds the zero sum $W$ of Section 4. It is therefore enough to show that every feasible pair has value below 1.

Relaxations of a near row

Fix a row $k$ and write $d=d_k/S$. The near-row constraint says $Q_k(\tau)\le1$, where

with $c_j\ge0$ collecting the coefficients $x_c/(D_{ck}/S)$ and $n/(D/S)$ and $u_j$ the features. The function $Q_k$ is convex and continuously differentiable, with $Q_k'(\tau)=-2\sum_j c_j(u_j-\tau)_++2\tau/d$. In the next two lemmas, $d>0$, all $c_j\ge0$, and $0\le a\le b$.

Lemma 10.2 (First-order relaxation). If $a\le\tau\le b$ and $Q_k(\tau)\le1$, then

Proof. Each $(u_j-\tau)_+^2\ge(u_j-b)_+^2$ since $\tau\le b$, and $\tau^2\ge a^2$ since $0\le a\le\tau$. $\square$

Lemma 10.3 (Tangent relaxation). Let $m=(a+b)/2$ and $h=(b-a)/2$. If $a\leq\tau\leq b$ and $Q_k(\tau)\leq1$, then at least one of the following holds:

Proof. By convexity, $1\geq Q_k(\tau)\geq Q_k(m)+Q'_k(m)(\tau-m)\geq Q_k(m)-h|Q'_k(m)|$, so $Q_k(m)-hQ'_k(m)\leq1$ or $Q_k(m)+hQ'_k(m)\leq1$. Expanding with $(u-m)_{+}^{2}\pm2h(u-m)_{+}=((u-m)_{+}\pm h)^2-h^2$ and $m^2\mp2hm=(m\mp h)^2-h^2$ gives the two cases. In case 1 a term with $u_j\leq m$ is 0. $\square$

The left-hand sides are linear in the $c_j$, hence in the column values $x$. In case 1 the coefficients $\widetilde{\theta}_j$ can be negative.

Certificates

Thresholds are measured in ticks of $1/T_S$, where $T_S=10^6$, and dual multipliers are scaled by $D_S=10^{12}$. In this subsection $[a,b]$ denotes an interval of thresholds in ticks, not a first-zero cell, and $m,h,p$ and $\beta$ are local to the relaxation formulas.

Definition 10.4 (Root threshold box). For row $k$, put

The root box is $\prod_k[0,e_k]$ in integer-tick coordinates. Its corresponding domain for real thresholds is $\prod_k[0,e_k/T_S]$.

Since $e_k^2S\geq d_kT_S^2$, every threshold with $\tau_k^2\leq d_k/S$ lies in $[0,e_k/T_S]$.

Definition 10.5 (Integer costs and budgets). Let $[a,b]$ be an interval of integer ticks with $0\leq a<b$, and let $v$ and $D>0$ be a stored feature and diagonal. The following costs are coefficients of column variables; the family terms are subtracted from the row budget.

(a) (First order.) The cost is $\left\lfloor(v-bS/T_S)_{+}^{2}/D\right\rfloor$. The budget of row $k$ is

(b) (Tangent, case $\iota\in\{0,1\}$.) Put $m=(a+b)/(2T_S)$, $h=(b-a)/(2T_S)$ and $\bar v=v-mS$ ($mS$ and $2hS$ are integers, since $2T_S$ divides $S$). The cost is $\left\lfloor S\Theta/D\right\rfloor$, where $\Theta=0$ if $\bar v\leq0$, and otherwise $\Theta=\left\lfloor\bar v(\bar v+2hS)/S\right\rfloor$ for $\iota=0$ and $\Theta=\left\lfloor\bar v(\bar v-2hS)/S\right\rfloor$ for $\iota=1$. The budget of row $k$ is $\left\lceil S\beta\right\rceil$, where $\beta$ is the exact rational number

with $p=a/T_S$ for $\iota=0$ and $p=b/T_S$ for $\iota=1$, and $\Theta_0(u)=((u-m)_{+}+h)^2-h^2$, $\Theta_1(u)=((u-m)-h)^2-h^2$ if $u>m$ and 0 otherwise.

The costs are the exact relaxation coefficients, multiplied by $S$ and rounded down. The budgets are at least the exact right-hand sides multiplied by $S$: the tangent budget is rounded up, and in the first-order budget each subtracted term is rounded down.

Definition 10.6 (Certificate). A certificate for a leaf is a finite binary tree starting from the root box of Definition 10.4. An internal node splits one coordinate $k$ of its box $[a,b]$ at an integer tick $a<\mathrm{mid}<b$, into $[a,\mathrm{mid}]$ and $[\mathrm{mid},b]$. At a terminal box, each row uses either the first-order relaxation or the two alternatives of the tangent relaxation. A relaxation case chooses one alternative for each tangent row. All combinations must be certified, because the alternative supplied by Lemma 10.3 can depend on the feasible point.

For consistency with the stored data, the mode is encoded by $o_k=2$ for a first-order row and $o_k=1$ for a tangent row. The case vector has $\iota_k=2$ when $o_k=2$, and $\iota_k\in\{0,1\}$ when $o_k=1$. In the stored trees the halves of a split are listed in the order $[a,\mathrm{mid}]$, $[\mathrm{mid},b]$, the rows are numbered from 0, and the relaxation cases of a terminal box are listed in lexicographic order of $(\iota_1,\ldots,\iota_K)$. For every relaxation case the node records either

(i) an exclusion: a row $k$ whose costs are all $\ge0$ and whose budget is $<0$; or

(ii) integer duals $Y_F,Y_C,Y_N,Y_E^+,Y_E^-\ge0$ and $Z_1,\ldots,Z_K\ge0$ such that for every column $c$

The value of the relaxation case is then

The value of a terminal box is the maximum of $-1$ and the values of its relaxation cases that carry duals (so it is $-1$ if every relaxation case is excluded), and the certificate is accepted if every terminal box has value $<S$. The data must be well formed: all diagonals and correlation terms are positive, and all family counts are nonnegative.

Theorem 10.7 (Soundness of the integer certificates). Let a leaf have positive diagonals and correlation terms and nonnegative family counts. If it has an accepted certificate in the sense of Definition 10.6, then every feasible pair $(x,\tau)$ of Definition 10.1 satisfies $\mathcal{V}(x)<1$. This conclusion holds for all real nonnegative column values, including fractional ones.

Proof. Let $(x,\tau)$ be feasible and put $t_k=\tau_kT_S$. Since $\tau_k^2\le d_k/S$ we have $t_k\le e_k$, so $t$ lies in the root box. At every internal node $t$ lies in one of the two halves, so there is a terminal box $\prod_k[a_k,b_k]$ that contains $t$.

For each row, Lemma 10.2 (if $o_k=2$) or Lemma 10.3 (if $o_k=1$) gives a case $\iota_k$ in which the exact relaxed inequality holds at $x$. Take the relaxation case $\iota$ so obtained. Multiply the relaxed inequality of row $k$ by $S$. Each coefficient of $x_c$ is then at least $\mathrm{cost}_{ck}$ and the right side is at most $\mathrm{budget}_k$, by the rounding directions in Definition 10.5 and since $x_c\ge0$ and $n\ge0$. Therefore

Here we used $S\cdot(v/S-b/T_S)_+^2/(D/S)=(v-bS/T_S)_+^2/D$ and the corresponding identities for the tangent terms.

If the relaxation case $\iota$ is excluded through row $k$, then the left side of (19) is $\ge0$ and the right side is $<0$, a contradiction. So $\iota$ carries duals. Multiply (18) by $x_c\ge0$ and sum over $c$. Then use the far, count and hidden-count constraints, with multipliers $Y_F,Y_C,Y_N\ge0$, the second-family equality, with the multiplier $Y_E^+-Y_E^-$ of either sign, and (19), with multipliers $Z_k\ge0$. This gives

Hence first $+$ final $+ \sum_{c} G_{c}x_{c}$ is at most the value of $\iota$, which is at most the value of the terminal box, which is $< S$. $\square$

Remark 10.8. The formal proof of Theorem 10.7 brought out that the family counts $n$ must be nonnegative. With a negative count, the first-order relaxation can fail. The leaf builders already ensure $n \ge0$, and the checker requires it.

The case tree

This section describes the finite case analysis of the middle range $0.1 \le\lambda_{1} < 1.5$ and proves that it covers every configuration of zeros (Theorem 11.7). The analytic facts it uses are the zero-location results of Section 7, the costs of Section 5, the far budget of Section 6 and the near rows of Section 9. There are two tasks: showing that the cases cover every possible configuration, and showing that a configuration in a case gives a feasible point of its program. These are proved separately before being combined in Theorem 11.7. Table 10, after Section 11.4, lists the levels of the analysis, with their numbers and the places where they are defined and checked.

Configurations.

Definition 11.1 (Zero configuration). Fix a sufficiently large $q$. Its zero configuration is the multiset $\mathcal{Z}$ of pairs $(\chi,\rho)$ with $\chi\ne\chi_{0}$ a character modulo $q$, $\rho\in R(l)$ and $L(\rho,\chi)=0$, each pair repeated according to the multiplicity of $\rho$.

Minima over empty sets are $+\infty$. In the middle range the first zero exists. The quantities of Section 2 are read off from $\mathcal{Z}$: the first zero $(\chi_{1},\rho_{1})$, the first family $\mathcal{F}_{1}$ with $n=|\mathcal{F}_{1}|$, the parameters $\lambda_{2},\lambda_{3}$ and the family $\mathcal{F}_{2}$ (Definition 2.1), and the distinguished occurrences, $\lambda'$ and an admissible $\rho'$ (Definition 2.2); for type rc, $\mu_{1}>0$. In addition we use the following.

• The height-one parameter of $\chi$ is $\nu(\chi)=\min\{\lambda_{\rho}:(\chi,\rho)\in\mathcal{Z},\ |\gamma_{\rho}|\le1\}$, and the $T^{*}$-parameter is $\nu_{T^{*}}(\chi)$, defined in the same way with $|\gamma_{\rho}|\le T^{*}$ (Definition 6.4). Since $L(\rho,\chi)=0$ if and only if $L(\bar{\rho},\bar{\chi})=0$, both are functions of the family. We put $\nu_{*}=\min\{\nu(\mathcal{F}):\mathcal{F}\ne\mathcal{F}_{1}\}$.

• The configuration is inside if $\rho_{1}\in R_{B}$ and outside otherwise. Since $\lambda_{1}<1.5<C_{B}$ in the middle range, outside means $|\mu_{1}|>C_{B}$. Type rr is always inside.

The definitions give $\lambda_{1}\le\lambda'$, $\lambda_{1}\le\lambda_{2}\le\lambda_{3}$, and $\nu_{T^{*}}(\chi)\ge\nu(\chi)\ge\lambda_{2}$ for $\chi\notin\mathcal{F}_{1}$.

Specifications.

A specification records interval information about a configuration. It may describe no actual configuration; in that event the corresponding case can be excluded. Its two lower bounds $l_{2}$ and $r$ have different scopes: $l_{2}$ applies throughout $R(l)$, whereas $r$ applies only to the height-one representatives.

Definition 11.2 (Specification). A specification is a tuple

with the following data:

• a type $\mathrm{t}\in\{\mathrm{rr},\mathrm{rc},\mathrm{complex}\}$ and a first-zero cell $[a,b]$, where $a<b$;

• a lower bound $p$ for $\lambda'$, and optionally an interval $\mathrm{gap}=[\mathrm{lo}',\mathrm{hi}']$ with $p\le\mathrm{lo}'<\mathrm{hi}'\le\infty$, called a gap (used only for type complex);

• a global second-family lower bound $l_{2}$ and a height-one lower bound $r$, with $l_{2}\le r\le3$;

• either no reservation, or a pair $\mathrm{res}=(\mathrm{hi}_{2},n_{2})$ with $r<\mathrm{hi}_{2}\le2$ and $n_{2}\in\{1,2\}$; in the latter case put $\mathrm{lo}_{2}=r$;

• a height class $\mathrm{loc}\in\{\mathrm{inside},\mathrm{outside}\}$.

The configuration set $C(\mathfrak{s})$ consists of the configurations for which some admissible choices of the first zero and the minimizing families satisfy all of the following:

• (C1) the type is $t$ and $\lambda_{1}\in[a,b]$;

• (C2) $\lambda'\ge p$, and $\lambda'\in[\mathrm{lo}',\mathrm{hi}']$ if a gap is present (with $\lambda'=\infty$ allowed when $\mathrm{hi}'=\infty$);

• (C3) $\lambda_{2}\ge l_{2}$;

• (C4) if $\mathrm{res}$ is none, then $\nu(\mathcal{F})\ge r$ for every family $\mathcal{F}\ne\mathcal{F}_{1}$; if $\mathrm{res}=(\mathrm{hi}_{2},n_{2})$, then $\nu_{*}\in[r,\mathrm{hi}_{2}]$ and some family $\mathcal{F}_{\mathrm{res}}\ne\mathcal{F}_{1}$ with $\nu(\mathcal{F}_{\mathrm{res}})=\nu_{*}$ has exactly $n_{2}$ characters (so it is a real character if $n_{2}=1$ and a pair of conjugate nonreal characters if $n_{2}=2$);

• (C5) the configuration is inside if $\mathrm{loc}=\mathrm{inside}$ and outside if $\mathrm{loc}=\mathrm{outside}$.

In the reserved case, (C4) implies $\nu(\mathcal{F})\ge r$ for every $\mathcal{F}\ne\mathcal{F}_{1}$. We call $\mathcal{F}_{\mathrm{res}}$ the reserved family and its characters reserved; the characters outside $\mathcal{F}_{1}$ and $\mathcal{F}_{\mathrm{res}}$ are ordinary.

In a formula containing $\mathrm{lo}'$, that argument is omitted if no gap is present. Likewise, reserved-family terms are absent when there is no reservation.

The reserved family is a family with the least height-one parameter among the families other than the first; it is not assumed to be the family $\mathcal{F}_{2}$ that attains $\lambda_{2}$, which may have its nearest zero at a height above 1. The following three facts are used in every leaf.

Lemma 11.3 (Consequences of reserving a family). Let $\mathfrak{s}$ be a specification and fix a configuration and admissible choices witnessing its membership in $C(\mathfrak{s})$.

(i) (Count.) At most two characters $\chi\notin\mathcal{F}_{1}$ have $\nu_{T^{*}}(\chi)<\lambda_{3}$.

(ii) (Drop.) If $\mathfrak{s}$ is reserved, then $\nu_{T^{*}}(\chi)\ge\nu(\chi)\ge\lambda_{3}$ for every ordinary character $\chi$.

(iii) (Cap.) If $\mathfrak{s}$ is reserved, then $\lambda_{2}\le\nu_{*}\le\mathrm{hi}_{2}$.

Proof. (i) Such a character has a zero in $R(l)$ with parameter below $\lambda_{3}$, so it lies in $\mathcal{F}_{2}$, and $|\mathcal{F}_{2}|\le2$. (ii) Let $\mathcal{F}$ be the family of an ordinary character. If $\mathcal{F}\ne\mathcal{F}_{2}$, every zero of $\mathcal{F}$ in $R(l)$ has parameter at least $\lambda_{3}$. If $\mathcal{F}=\mathcal{F}_{2}$, then $\mathcal{F}_{\mathrm{res}}\notin\{\mathcal{F}_{1},\mathcal{F}_{2}\}$, so $\nu(\mathcal{F}_{\mathrm{res}})\ge\lambda_{3}$, and $\nu(\mathcal{F})\ge\nu_{*}=\nu(\mathcal{F}_{\mathrm{res}})$ by minimality. (iii) The height-one representative of $\mathcal{F}_{\mathrm{res}}$ is a zero in $R(l)$ of a character outside $\mathcal{F}_{1}$. $\square$

The source cover

The roots of the case tree are 2768 specifications, each used with the height class inside, and the 1685 of them whose type is not rr used also with the height class outside: 4453 roots in all. They are constructed from the zero-location results of Section 7 in four steps.

1. Parent rows. There are 58 parent rows $(t,A_{j},B_{j},p_{j},r_{j})$: 33 of type rr tiling $[0.1,1.5]$, 13 of type rc tiling $[0.628,1.5]$, and 12 of type complex tiling $[0.44,1.5]$. Each is the statement

which is proved in Proposition 7.5 from the tables of Heath-Brown and Xylouris.

2. Base cells. Each parent interval is cut into cells of width 0.01, giving 340 base cells.

3. Cells and gaps. The base cells are tiled by 478 first-zero cells in all. A cell $[\mathrm{lo},\mathrm{hi}]$ of the parent $j$ receives numbers $p\le\max(p_{j},\mathrm{lo})$ and $l_{2}\le\max(r_{j},\mathrm{lo})$; this is valid since $\lambda',\lambda_{2}\ge\lambda_{1}\ge\mathrm{lo}$. A cell of type complex may be partitioned by gaps $[p,m_{1}],[m_{1},m_{2}],\ldots,[m_{k},\infty]$ for $\lambda'$. The cells that are not partitioned, and the gaps of those that are, are the 684 gap cases.

4. Records. Each gap case is either unreserved, with one specification with $r=l_{2}$ and no reservation, or reserved with a chain $l_{2}=u_{0}<u_{1}<\cdots<u_{m}=e\le2$, with specifications $(\mathrm{res}=(u_{j+1},n_{2}),\,r=u_{j})$ for $n_{2}=1,2$ and $0\le j<m$, followed by an unreserved tail specification with $r=e$.

By type there are 1083 specifications of type rr (195 unreserved, 444 reserved with $n_{2}=1$ and 444 with $n_{2}=2$), 115 of type rc (all unreserved and without gap) and 1570 of type complex.

Theorem 11.4 (Coverage by the root cases). Every configuration with $0.1 \le\lambda_{1}<1.5$ lies in $C(s)$ for at least one of the 4453 root specifications $s$.

Proof. Let t be the type. By Proposition 7.1, $\lambda_{1}>0.628$ if $\mathrm{t}=\mathrm{rc}$ and $\lambda_{1}>0.44$ if $\mathrm{t}=\mathrm{complex}$, so $\lambda_{1}$ lies in some cell of type t (the cells are closed, and a boundary value lies in two cells). By [T2] for the parent of the cell, $\lambda' \ge p$ and $\lambda_{2} \ge l_{2}$. If the cell is partitioned, some gap contains $\lambda'$. In the corresponding gap case, if it is unreserved, every family $\mathcal{F} \ne\mathcal{F}_{1}$ has $\nu(\mathcal{F}) \ge\lambda_{2} \ge l_{2}=r$. If it is reserved with chain $u_{0}<\cdots<u_{m}$, then $\nu_{*} \ge\lambda_{2} \ge u_{0}$; if $\nu_{*}<e$ we choose $j$ with $\nu_{*} \in[u_{j},u_{j+1}]$ and a minimizing family, which is a real character or a nonreal pair, and the configuration lies in the corresponding record; otherwise it lies in the tail record. Finally the height class is determined by $|\mu_{1}|$. □

Refinement trees

Each root is the root of a finite refinement tree. Every internal node carries a specification (computed from the root, never read from the data) and is of one of the following kinds. In each case every configuration of the node lies in the configuration set of one of its children, or the node has no configuration at all.

• First-zero split at $a<m<b$: children with cells $[a,m]$ and $[m,b]$, all other data kept. Exhaustive, since $\lambda_{1} \le m$ or $\lambda_{1} \ge m$.

• Second-family split at $r<m<\mathrm{hi}_{2}$ (reserved nodes): children $(\mathrm{hi}_{2} \coloneqq m)$ and $(r \coloneqq m,\ \mathrm{lo}_{2} \coloneqq m)$. Exhaustive, since $\nu_{*} \le m$ or $\nu_{*} \ge m$, and in the second case every family $\mathcal{F} \ne\mathcal{F}_{1}$ has $\nu(\mathcal{F}) \ge\nu_{*} \ge m$.

• Gap split at $\mathrm{lo}'<m<\mathrm{hi}'$ (type complex): children with gaps $[\mathrm{lo}',m]$ and $[m,\mathrm{hi}']$. Exhaustive.

• Location update: a lower bound $\lambda_{2}>h$ valid on the whole cell (Propositions 7.10 and 7.12; Lemma 7.15) replaces $l_{2}$ and $r$ by $\max(l_{2},h)$ and $\max(r,h)$, and excludes the node if it is reserved with $\mathrm{hi}_{2} \le h$. Sound, since every family $\mathcal{F} \ne\mathcal{F}_{1}$ has $\nu(\mathcal{F}) \ge\lambda_{2}$.

• Exclusion of a reserved interval or a gap by a lower bound for $\lambda_{2}$ or $\lambda'$ (Propositions 7.11 and 7.13), or of a whole node by the positivity argument of Proposition 7.14.

Before a leaf model is built, all applicable location updates are applied once more to its specification; this only adds valid implications. None of these nodes depends on $L$. The terminal nodes are the leaves. The inside trees have 3146 leaves. Of these, 233 (in 199 roots) are stored in a separate file and are called supplementary leaves. Every leaf, supplementary or not, receives the same model. The outside trees have 1642 leaves. In all there are 4788 leaves; 343 inside roots and 70 outside roots have none, because all their configurations are excluded.

The leaf model

This subsection gives the real coefficients before integer scaling. An interval $[\mathrm{lo},\mathrm{hi})$ is charged at its left endpoint in the objective, where the cost is largest, and at its right endpoint in density constraints, where the available weight is smallest. These choices enlarge the feasible set and give an upper bound for every actual distribution of characters in the interval.

Let $\mathfrak{s}$ be the specification of a leaf, after the location updates. Let $s$ denote the shift $\min(1.9,p^{*},l_{2})$, where $p^{*}=\mathrm{lo}'$ if $\mathfrak{s}$ has a gap and $p^{*}=p$ otherwise, and let $\lambda_{3}^{\mathrm{lo}}$ be the lower bound for $\lambda_{3}$ of Proposition 7.16, computed from $[a,b]$, the type and, for a reserved $\mathfrak{s}$, the cap $\mathrm{hi}_{2}$. The leaf program (Definition 10.1) has the columns of Table 11. Every number is rounded in the safe direction at scale $S$: far weights, features and the hidden-tail count coefficient $1/w(\mathrm{end})$ down; objective coefficients, diagonals, correlation terms, the far budget and the constant first up. Let $O$ be the set of ordinary characters with a zero in $R_{P}$, and $\mathcal{H}$ the set of first-family characters with a zero in $R_{P}$.

Column	Value $x_{c}$	$G_{c}$	$W_{c}$	$C_{c}$	$N_{c}$	$E_{c}$	Near rows
ordinary bin $[\xi_{i},\xi_{i+1})$	number of $\chi\in O$ with $\nu_{T^{*}}(\chi)$ in the bin	$\mathcal{G}(\xi_{i})$	$w(\xi_{i+1})$	0 or $1^{\mathrm{a}}$	0	0	entry (a)
tail	$\sum_{\substack{\chi\in O\ \mathrm{with}\\ \nu_{T^{*}}(\chi)\geq R}}w(\nu_{T^{*}}(\chi))$	$\mathcal{G}(R)/w(R)$	1	0	0	0	omitted
reserved column$^{\mathrm{b}}$ $[\mathrm{lo}_{2,j},\mathrm{hi}_{2,j}]$	$z_{j}=n_{2}$ if $\nu_{*}$ is in the column, else 0	$\mathcal{G}_{\phi_{2}}(\mathrm{lo}_{2,j})$	$w(\mathrm{hi}_{2,j})$	0	0	1	entry (f)
hidden bin$^{\mathrm{c}}$ $[\mathrm{lo}_{h},\mathrm{hi}_{h})$	number of $\chi\in\mathcal{H}$ with $t_{\chi}$ in the bin	$e^{-A_{\mathrm{lo}_{h}}B_{\phi_{1}}(p^{\circ})}$	$w(\mathrm{hi}_{h})$	0	1	0	entry (g)
hidden tail$^{\mathrm{c}}$	$\sum w(t_{\chi})$ over $\chi\in\mathcal{H}$ with $t_{\chi}\geq\mathrm{end}$	$\dfrac{e^{-A_{\mathrm{end}}B_{\phi_{1}}(p^{\circ})}}{w(\mathrm{end})}$	1	0	$\dfrac{1}{w(\mathrm{end})}$	0	omitted

(a) Ordinary bins. Let $R=\max(3,r)$ and let $r=\xi_{0}<\xi_{1}<\cdots<\xi_{N}=R$ consist of $r$, $R$ and the multiples of $1/\mathrm{den}$ between them, where $\mathrm{den}\in\{200,400,500,800,2000\}$ is fixed per leaf. The bins are the intervals $[\xi_{i},\xi_{i+1})$, except that in a reserved leaf the bins with $\xi_{i+1}\leq\lambda_{3}^{\mathrm{lo}}$ are omitted. The ordinary characters beyond $R$ form the tail.

(b) Reserved family. Either (fixed terms) the family is charged $n_{2}\mathcal{G}_{\phi_{2}}(\mathrm{lo}_{2})$ in first and $n_{2}w(\mathrm{hi}_{2})$ in the far budget, and it appears in the near rows as a family term with its feature at $\mathrm{hi}_{2}$ (there are then no second-family columns, and the count $n_{E}$ of the second-family equality of Definition 10.1 is 0); or (columns) the interval $[\mathrm{lo}_{2},\mathrm{hi}_{2}]$ is cut at the multiples of $1/400$ into reserved columns $[\mathrm{lo}_{2,j},\mathrm{hi}_{2,j}]$, with values $z_{j}$ and $\sum_{j}z_{j}=n_{2}=n_{E}$.

(c) Hidden columns (outside leaves only). Put $p^{\circ}=\max(a,p,\mathrm{lo}')$ and $\mathrm{end}=\max(3,p^{\circ})$. The hidden bins $[\mathrm{lo}_{h},\mathrm{hi}_{h})$ divide $[p^{\circ},\mathrm{end}]$ on the grid $1/\mathrm{den}$, and the hidden tail collects the values $t_{\chi}\geq\mathrm{end}$. The hidden count is at most $n$.

Before scaling and rounding, the far budget is $(1+\eta)V-\mathbf{1}_{\mathrm{inside}}n\,w(b)$, with a further subtraction of $n_{2}w(\mathrm{hi}_{2})$ for a fixed reserved-family term. The stored integer budget $F$ is formed by rounding the initial budget up and each subtraction down, as in Section 6. The count budget is 2, and the stored allowance satisfies $\mathrm{final}/S=5\eta$. For an inside leaf, $\mathrm{first}/S$ is rounded up from $J_{1}=\min(J_{\mathrm{old}},J_{\mathrm{new}})$ with anchor $p^{*}$, omitting $J_{\mathrm{new}}$ when $p^{*}<b$. For an outside leaf it starts at 0. The fixed reserved-family charge, if used, is added with upward rounding. The near rows are those of Section 9.

Subdivision of a leaf

A leaf may be certified through finitely many subcases, each of which is certified with its own leaf model. Besides the second-family split above, three kinds are used. The first replaces the specification by three specifications. The other two restrict one further quantity $Q$ of the configuration to one of finitely many intervals $I$ whose union is the range of $Q$; the configuration set of the subcase $(\mathfrak{s},I)$ is the set $C(\mathfrak{s},I)$ of configurations for which, for some admissible choice of $(\chi_{1},\rho_{1})$, of the minimizers and of $\rho'$, (C1)–(C5) hold and $Q\in I$. Then $C(\mathfrak{s})$ is the union of the sets $C(\mathfrak{s},I)$, and all the results of Sections 9 and 11 that are stated for $C(\mathfrak{s})$ hold for $C(\mathfrak{s},I)$ with the same proofs.

Lemma 11.5 (Reservation split). Let $\mathfrak{s}$ be unreserved with ordinary lower bound $r$, and let $r<h\le2$. Let $\mathfrak{s}_{1}$ and $\mathfrak{s}_{2}$ be $\mathfrak{s}$ with the reservation $(h,1)$ and $(h,2)$ (so $\mathrm{lo}_{2}=r$ and $\mathrm{hi}_{2}=h$), and let $\mathfrak{s}_{0}$ be $\mathfrak{s}$ with $r$ replaced by $h$. Then $C(\mathfrak{s})\subseteq C(\mathfrak{s}_{0})\cup C(\mathfrak{s}_{1})\cup C(\mathfrak{s}_{2})$.

Proof. If $\nu_{*}<h$, a family $\mathcal{F}\ne\mathcal{F}_{1}$ attaining $\nu_{*}$ exists, and it is a real character or a pair of conjugate nonreal characters; the configuration lies in $C(\mathfrak{s}_{1})$ or $C(\mathfrak{s}_{2})$, since $\nu_{*}\ge r$. Otherwise every family $\mathcal{F}\ne\mathcal{F}_{1}$ has $\nu(\mathcal{F})\ge h$, and the configuration lies in $C(\mathfrak{s}_{0})$. $\square$

In a reserved child the model uses only the minimality of the reserved family (Lemma 11.3(ii)) and the cap $\lambda_{2}\le h$ (Lemma 11.3(iii)); it never uses that the reserved family is $\mathcal{F}_{2}$. The split of Lemma 11.5 is used with $h=\min(\lambda_{3}^{\mathrm{lo}},2)$ on four inside roots and on five roots that contain supplementary leaves.

Height split for type $\mathrm{rc}$. An inside leaf of type $\mathrm{rc}$ is split by $Q=\mu_{1}$ (recall that $\mu_{1}>0$ for this type) into $\mu_{1}\in[0,1]$ and $\mu_{1}\in[1,\infty)$, which changes only the family row (Section 9).

Second-zero height split. An inside leaf of type complex with a finite gap is split by $Q=|y|$, where $y=\mu_{1}-\mu_{\rho'}=(\gamma_{1}-\gamma')L$ is the normalized height difference between $\rho_{1}$ and the nondistinguished zero $\rho'$ of $\chi_{1}$ realizing $\lambda'$ (a zero of $\chi_{1}$, not of $\overline{\chi}_{1}$; Section 2), into six pieces, according as $|y|$ lies in $[0,1]$, $[1,\frac{3}{2}]$, $[\frac{3}{2},2]$, $[2,3]$, $[3,5]$ or $[5,\infty)$. This changes only the family row. A piece may be certified by an exclusion (Definition 10.6(i)) rather than by duals: the near row alone is then infeasible.

Realization and the cover theorem

Proposition 11.6 (Realization). Let $\mathfrak{s}$ specify a leaf or one of its subcases, with the program constructed in Section 11.5. For every sufficiently large $q$ whose configuration lies in $C(\mathfrak{s})$, there is a feasible pair $(x,\tau)$ such that

The column values $x$ are obtained from the actual character counts and weighted tail masses. The threshold $q_{0}$ may be chosen uniformly over the finite set of leaves and subcases.

Proof. Let $O$ and $\mathcal{H}$ be as in Section 11.5 ($\mathcal{H}$ is used only if the configuration is outside). For $\chi\in O$ we have $\nu_{T^{*}}(\chi)\le C_{P}$ and $\nu_{T^{*}}(\chi)\ge\nu(\chi)\ge r$ by (C4). Put $x_{i}$ equal to the number of $\chi\in O$ with $\nu_{T^{*}}(\chi)\in[\xi_{i},\xi_{i+1})$, $x_{\mathrm{tail}}=\sum w(\nu_{T^{*}}(\chi))$ over $\chi\in O$ with $\nu_{T^{*}}(\chi)\ge R$, $z_{j}$ equal to $n_{2}$ for the reserved column containing $\nu_{*}$ and 0 otherwise, $x_{i}^{h}$ equal to the number of $\chi\in\mathcal{H}$ whose value $t_{\chi}$ (Section 5.7) lies in the hidden bin $i$, and $x_{\mathrm{tail}}^{h}=\sum w(t_{\chi})$ over the $\chi\in\mathcal{H}$ in the hidden tail. In a reserved leaf no $\chi\in\mathcal{O}$ lies in an omitted bin, by Lemma 11.3(ii) and $\lambda_{3}\geq\lambda_{3}^{\mathrm{o}}$.

Far. The characters of $\mathcal{O}$, the reserved characters, the first-family characters of $\mathcal{H}$ and, for an inside configuration, $\chi_{1}$ and $\overline{\chi}_{1}$ are distinct, and to each we assign one zero with $|\gamma|\leq1$ and parameter at most $CP$: the $T^{*}$-representative, the height-one representative, the zero realizing $t_{\chi}$, and $\rho_{1}$ or $\overline{\rho}_{1}$, which has $|\gamma_{1}|\leq C_{B}/\mathcal{L}\leq1$. By (6.1), and since $w$ is decreasing, the far constraint holds.

Count. By Lemma 11.3(i), since a character in a counted bin has $\nu_{T^{*}}(\chi)<\xi_{i+1}\leq\lambda_{3}^{\mathrm{o}}\leq\lambda_{3}$.

Hidden count and second family. We have $|\mathcal{H}|\leq n$, a character in the hidden tail contributes $w(t_{\chi})/w(\mathrm{end})\leq1$ to the hidden count, and $\sum_{j}z_{j}=n_{2}=n_{E}$ when the reserved family is charged through columns (otherwise there are no second-family columns and $n_{E}=0$).

Near rows. By Proposition 9.7, each near row holds for some threshold, with features at least and diagonals at most those of the columns.

Value. We use (5.2). An ordinary character in a bin costs at most the objective $\mathcal{G}(\xi_{i})$ at the left end of the bin, a reserved character in a column at most $\mathcal{G}_{\phi_{2}}$ at the left end (with fixed terms, at most $\mathcal{G}_{\phi_{2}}(\mathrm{lo}_{2})$), and the tail characters at most $\mathcal{G}(R)/w(R)$ per unit of far mass, by the monotonicity of $\mathcal{G}$, $\mathcal{G}_{\phi_{2}}$ and $\mathcal{G}/w$ (Lemma 5.1(v)). For an outside configuration we apply Proposition 5.9 with the anchor $p^{\circ}=\max(a,p,\mathrm{lo}^{\prime})$, which is admissible because every non-distinguished occurrence of the first family has parameter at least $\lambda^{\prime}\geq p^{\circ}$; then $t_{\chi}\geq p^{\circ}$, a character in a hidden bin $[\mathrm{lo}_{h},\mathrm{hi}_{h})$ costs at most the objective $e^{-A\mathrm{lo}_{h}}B_{\phi_{1}}(p^{\circ})$ since $e^{-A\lambda}$ is decreasing, and a character in the hidden tail costs at most $e^{-A\cdot\mathrm{end}}B_{\phi_{1}}(p^{\circ})/w(\mathrm{end})$ per unit of far mass, since $e^{-A\lambda}/w(\lambda)$ is decreasing (Lemma 6.2). Including the directions of integer rounding, we obtain

Theorem 11.7 (The zero-sum bound in the middle range). For the case tree and certificates described in this paper, there is a threshold $q_{0}$ such that every $q\geq q_{0}$ with $0.1\leq\lambda_{1}<1.5$ satisfies

More precisely, its configuration belongs to a leaf or subcase whose program has a feasible pair $(x,\tau)$ with $W(q)+2\eta\leq\mathcal{V}(x)<1$.

Proof. By Theorem 11.4 the configuration lies in the configuration set of a root. Following the refinement tree and the subdivisions of Section 11.6, each of which is exhaustive, we reach a leaf or subcase whose configuration set contains the configuration; it is not an excluded node, because excluded nodes have no configurations. Proposition 11.6 gives the feasible pair, and Theorem 10.7 and the certificates of Section 14 give $\mathcal{V}(x)<1$. $\square$

The exterior regimes

The case tree covers $0.1\leq\lambda_{1}<1.5$. We now treat the remaining possibilities: an empty zero rectangle, a first zero with $\lambda_{1}<0.1$, and a first zero with $\lambda_{1}\geq1.5$. The small-parameter case is different from the others. Its first zero is real: it is an exceptional zero in the sense of Landau and Page [74, 97]. By the theorems of Siegel and Tatuzawa [118, 123] (see also [102, 103]) it cannot be extremely close to 1, but these bounds do not keep $\lambda_{1}$ away from 0. Although such a zero has a large cost, the Deuring–Heilbronn phenomenon [30, 53, 78] repels the remaining zeros. We use Heath-Brown’s quantitative estimates for this repulsion (see also [8] for explicit versions) and, for a sufficiently close zero, his separate theorem on Siegel zeros [50], by which a very strong exceptional zero is even helpful.

No zero in $R(l)$

If no nonprincipal $L(s,\chi)$ vanishes in $R(l)$, then for large $q$ none vanishes in $R_{P}$, because $C_{P}\le\frac{1}{3}\log\log\mathcal{L}$ and $C_{P}/\mathcal{L}\le l$. So $W=0$ and Proposition 4.4 gives a prime $p\equiv a\pmod q$ with $q^{A}<p<q^{L}$. Xylouris notes this case himself [145].

A tiny exceptional zero

Let $\eta_{\mathrm{HB}}(\delta)$ be the effectively computable constant of Input 3.10 and put

Proposition 12.1 (Primes in the presence of a very close real zero). Let $u_{\mathrm{exc}}$ be the positive cutoff defined above. For all sufficiently large $q$, if $\lambda_{1}<u_{\mathrm{exc}}$, then $P(a,q)\le q^{3.5}$ for every $(a,q)=1$.

Proof. By Proposition 7.1, $\chi_{1}$ and $\rho_{1}$ are real, so $\beta_{0}=\rho_{1}=1-\lambda_{1}/\mathcal{L}$ is a real zero of the real nonprincipal $L(s,\chi_{1})$. Since $\lambda_{1}<u_{\mathrm{exc}}\le\frac{1}{3}$ we have $\beta_{0}\ge1-1/(3\log q)$, and $((1-\beta_{0})\log q)^{-1}=1/\lambda_{1}>1/u_{\mathrm{exc}}\ge\eta_{\mathrm{HB}}(\frac{1}{2})$. Input 3.10 with $\delta=\frac{1}{2}$ gives the claim. $\square$

A small exceptional zero

For $u_{\mathrm{exc}}\le\lambda_{1}\le0.1$ we use a different prime weight, a single triangle, and Heath-Brown’s own estimates. The inputs are the following.

Input 12.2 (Prime detection with a single triangle [51]). Let $L'>2K+3$, let $f=h_{L',K}$ and $F(z)=e^{-(L'-2K)z}F_{2}(z)$ with $F_{2}(z)=\left(\frac{1-e^{-Kz}}{z}\right)^{2}$. For every $\varepsilon>0$ there is $R=R(\varepsilon)$ such that for large $q$

where $\sum'$ runs over the zeros with $1-R/\mathcal{L}\le\beta\le1$, $|\gamma|\le R/\mathcal{L}$.

Input 12.3 (Per-character bound for the triangular kernel [51]). Let $\chi\ne\chi_{0}$, $0<\lambda\le R$, and suppose that $L(s,\chi)$ has no zero with $1-\lambda/\mathcal{L}<\beta\le1$ and $|\gamma|\le1$. Then for every $\eta_{1}>0$ and $q\ge q(\eta_{1},\lambda)$

Input 12.4 (Heath-Brown’s weighted density estimate). In the notation of [51] and [145], the following holds. Let $\varepsilon,c_{1},c_{2}>0$ and $\phi=\max_{\chi}\varphi(\chi)$. For $q\ge q(\varepsilon,c_{1},c_{2})$ and any choice, for each nonprincipal $\chi$ with a zero in $\beta\ge1-\lambda_{0}/\mathcal{L}$, $|\gamma|\le1$ $(\lambda_{0}=\frac{1}{3}\log\log\mathcal{L})$, of one such zero with parameter $\lambda(\chi)$,

The left side decreases and the right side increases with $\phi$, so the inequality holds with $\phi=\frac{1}{3}$.

We also use Input 7.2(b),(d) and the following two rows of Heath-Brown’s tables, which hold for all $\lambda_{1}\le0.10$ bounded below by a fixed positive constant [51] (see the remark after Input 7.2; we use them for $\lambda_{1}\ge0.08$): if $\chi_{1}$ and $\rho_{1}$ are real and $\lambda_{1}\le0.10$, then $\lambda'\ge4.96-\varepsilon$ and $\lambda_{2}\ge2.83-\varepsilon$ for $q\ge q(\varepsilon)$. We have recomputed these rows from Heath-Brown’s printed parameters, obtaining $4.96355$ and $2.84577$; we use them with $\varepsilon=0.001$. (Table 5 is stated under $\lambda_{2}\le\lambda'$, which is redundant here since $\lambda'\ge4.959>2.83$.)

Fixed data. $L=3.99$, $K=0.1821$, $A_{s}=L-2K=3.6258$; $c_{1}=0.057$, $c_{2}=0.1554$, $\phi=\frac{1}{3}$;

$\alpha=1.09<\frac{12}{11}$, $B_1=K^2+\frac{K}{4}$, $m^*=\alpha\log12.5=2.7530442\ldots$, $m_2=2.829$ and $m'_2=4.959$. Let $H_s(u)=e^{-A_su}F_2(u)$ and $\mathcal{M}(u)=(K^2-H_s(u))/u$, and

Theorem 12.5 (The small exceptional-zero range). For all sufficiently large $q$ with $u_{\mathrm{exc}}\le\lambda_1\le0.1$, and every integer $a$ with $(a,q)=1$, there is a prime $p$ such that

The threshold is uniform over the fixed interval $[u_{\mathrm{exc}},0.1]$.

Proof. Write $u=\lambda_1$. By Proposition 7.1, $\chi_1$ and $\rho_1$ are real. The zero $\rho_1$ is simple, since a double zero would give $\lambda'=u$, whereas $\lambda'\ge(2-\varepsilon)\log(1/u)>u$ (Input 7.2(b)) on $[u_{\mathrm{exc}},0.08]$ and $\lambda'\ge4.959>u$ on $[0.08,0.1]$.

The criterion. Apply Input 12.2 with $L'=L$, since $3.99>2K+3$; the zero $\rho_1$ contributes $H_s(u)$. Enlarging $R$ only adds nonnegative terms to $\sum'$, so the inequality of Input 12.2 remains true with $R$ replaced by $\max(R,3)$; we use it with $R\ge3$, so that Input 12.3 applies with the values $\lambda\le m_2<3$ below. It remains to bound the other terms. Put $m(u)=\alpha\log(1/u)$ on $P_1=[u_{\mathrm{exc}},0.08]$ and $m(u)=m_2$ on $P_2=[0.08,0.1]$, and $m_1(u)=m(u)$ on $P_1$, $m_1(u)=m'_2$ on $P_2$. By Input 7.2(b),(d) (with $\varepsilon=\frac{12}{11}-\alpha$) and the two table rows, every zero $\rho\ne\rho_1$ of $L(s,\chi_1)$ in $R(l)$ has $\lambda_\rho\ge m_1(u)$, and every zero of every other nonprincipal $L(s,\chi)$ in $R(l)$ has $\lambda_\rho\ge m(u)$.

The first character. $L(s,\chi_1)$ has no zero with $\lambda<u_{\mathrm{exc}}/2$ and $|\gamma|\le1$, so Input 12.3 with $\lambda=u_{\mathrm{exc}}/2$ and $\varphi(\chi_1)=\frac14$ gives $\sum'_\rho|F_2((1-\rho)\mathcal{L})|\le B_{1/4}(u_{\mathrm{exc}}/2)+\eta_1\le B_1+\eta_1$, as $B_\phi$ is decreasing in $\lambda$, with limit $\phi K+K^2$ as $\lambda\to0$. Hence the zeros of $\chi_1$ other than $\rho_1$ contribute

The other characters. For $\chi\ne\chi_0,\chi_1$ with a zero in the rectangle of Input 12.2, let $\lambda(\chi)$ be the parameter of its zero of least parameter with $|\gamma|\le1$; then $\lambda(\chi)\ge m(u)$. Apply Input 12.3 with the fixed value $\lambda=m^*$ on $P_1$ (note $m^*\le m(u)$ there) and $\lambda=m_2$ on $P_2$, and with $\varphi(\chi)\le\frac13$. Since every zero in the rectangle has parameter at least $\lambda(\chi)$, the zeros of $\chi$ contribute at most $e^{-A_s\lambda(\chi)}(B_{1/3}(\bar m)+\eta_1)$, with $\bar m=m^*$ or $m_2$. Since $A_s>a_d$, the function $e^{-A_st}(e^{a_dt}-e^{b_dt})/t=Q(t)$ is decreasing, so $e^{-A_s\lambda(\chi)}\le Q(m(u))\lambda(\chi)/(e^{a_d\lambda(\chi)}-e^{b_d\lambda(\chi)})$. By Input 12.4, the characters other than $\chi_1$ contribute

Monotonicity. We have $K^2-H_s(u)=\int h_{L,K}(t)(1-e^{-ut})\,\mathrm{d}t$, so $\mathcal{M}(u)=\int h_{L,K}(t)\frac{1-e^{-ut}}{u}\,\mathrm{d}t$, which decreases in $u$ because $h_{L,K}\ge0$ and $(1-e^{-ut})/u$ decreases for each $t>0$. On $P_1$ we have $u=e^{-m/\alpha}$, and

with $A_s-a_d-1/\alpha=\frac{236171}{327000}>0$; both increase with $u$. On $P_2$ the bounds for $T_1$ and $T_2$ do not depend on $u$, and $u\ge0.08$. Hence on $P_i$

where

and $\mathcal{E}=\eta_{1}(1+(V_{d}+\varepsilon_{2})(a_{d}-b_{d}))+\varepsilon_{2}B_{1/3}(0)(a_{d}-b_{d})$, using $Q\leq a_{d}-b_{d}$.

Numerical margins. The interval computations give the following values, displayed here to ten decimal places:

	$\mathcal{M}$	other characters	first character	$\varpi_{i}$
$\mathcal{P}_{1}$	0.1088458429	0.0783116746	0.0000454655	0.0304887029
$\mathcal{P}_{2}$	0.1050056698	0.0668874692	0.0000000153	0.0381181853

The certified lower bounds give $\varpi_{i}>0.0304$. Choose $\eta_{1}$, $\varepsilon_{2}$ and the $\varepsilon$ of Input 12.2, after $u_{\mathrm{exc}}$, with $\mathcal{E}+\varepsilon<0.03u_{\mathrm{exc}}$. Then the bracket in Input 12.2 is at least $(0.0304-0.03)u_{\mathrm{exc}}>0$, and there is a prime $p\equiv a\pmod{q}$ with $\log p/\mathcal{L}\in(A_{s},L)$. □

At $L=3.99$, the margins are approximately 28% and 36% of the main term. Exploratory evaluations with the same $K$, $c_{1}$, and $c_{2}$ remain positive at $L=3.90$; the certified exterior result used here is the one at 3.99.

A large first zero.

Theorem 12.6 (The large first-zero range). For all sufficiently large $q$ with a nonempty zero rectangle and $\lambda_{1}\geq1.5$,

Consequently, every reduced residue class contains a prime $p$ with $q^{3.156342}<p<q^{3.99}$.

Proof. Every nonprincipal character with a zero in $R_{P}$ is treated as ordinary, with its height-one representative, of parameter $\lambda_{\chi}\geq\lambda_{1}\geq1.5$; by Corollary 5.7 it contributes at most $\mathcal{G}(\lambda_{\chi})+\varepsilon_{1}e^{-A\lambda_{\chi}}$. Consider the leaf program with the bins $[1.5+\frac{j}{100},1.5+\frac{j+1}{100})$, $0\leq j<150$, and the tail $[3,\infty)$; the far budget $F=(1+\eta)V$ with profile I (Section 6); first $=0$ and final $=5\eta$; no count, hidden or second-family constraints; and one near row. The near row is Corollary 8.10 with $\Xi\equiv0$, $s=s_{1}=\frac{3}{2}$, detector $\hat{f}_{2}$ ($\gamma_{f}=1$), Gram test $\hat{f}_{3}$ ($\gamma_{g}=\frac{3}{2}$) and no family entries, normalized as in Section 9.2 with $t_{0}=1$ (so $I_{u}$ is the upper Riemann sum over 2000 cells of each of $[0,1)$ and $[1,2)$): every zero of every nonprincipal $L$-function in $\mathbf{R}(l)$ has parameter at least $1.5=s_{1}$, so the safe-anchor hypothesis holds, and every entry has no zero to the right of its anchor $\min(\operatorname{lo},s)=s$. The actual characters give a feasible point as in Proposition 11.6.

The certificate consists of 36 contiguous threshold intervals covering $[0,e]$; on each it uses the first-order relaxation and integer duals $Y_{F},Z\geq0$ satisfying (18); there are no exclusions and 5436 column inequalities. The largest value is 0.9966813270344512, on the threshold interval $[0.1625,0.165]$, where $Y_{F}=5145926349$ and $Z=1593393341176$. By Theorem 10.7, $\mathcal{V}(x)<0.9966814$, and $W\leq\mathcal{V}(x)-2\eta$ as in Proposition 11.6. Proposition 4.4 gives the prime. □

In this regime one near row, together with the far-density budget, suffices. All characters are treated by the same per-character bound.

Assembly of the proof

Each preceding argument holds once $q$ exceeds a threshold depending on fixed choices. We first verify that these choices can be made in a noncircular order. A single threshold then works for every leaf and every exterior case.

The order of the choices

The constants of the argument are chosen in the following order. Each stage depends only on the earlier ones, and there are finitely many choices at each stage.

(S1) The fixed data: the exponent $L=3.99$ and the kernel of Section 4; the two far profiles of Section 6; the test functions, anchors, sieve weights and grids of all near rows (Section 9); the case tree, its leaf programs and their certificates (Sections 10 and 11); the zero-location rows of Section 7; the data of the exterior regimes (Section 12); $\eta=\eta'=10^{-6}$, $\lambda_{11}=0.05$ and $s_{\max}=3$; and $u_{\mathrm{exc}}=\min(1/\eta_{\mathrm{HB}}(\frac{1}{2}),0.08)$, with $\eta_{\mathrm{HB}}(\delta)$ from Input 3.10; and the envelope tolerance $\varepsilon_{1}=\eta/23$ of Section 5.8.

(S2) The prime window: $C_{P}\geq\max(C_{0}(\eta H_{0}),3)$, with $C_{0}$ from Input 4.2.

(S3) The zero count $N=N(C_{P})$ of Lemma 5.3; the mesh $h_{\mathcal{A}}$ and the finite set of anchors $\mathcal{A}$ of Section 5.3; then the smoothing parameter $\theta$ of Section 5.2, chosen so that (7) holds for $\varepsilon_{1}=\eta/23$; then the constant $c_{\mathrm{out}}$ of Proposition 5.9.

(S4) The count $K_{0}$ of Section 6.3, a bound, uniform in $q$, for the number of zeros of all nonprincipal $L$-functions modulo $q$ with $\lambda\leq s_{\max}$ and $|\gamma|\leq2$ (Input 3.9). Then the separation $M$, so large that for $|Y|\geq M$ we have $|\operatorname{Re}F(x+iY)|\leq\eta_{\min}''/(2K_{0}+2)$ for every detector and $x\in[-s_{\max},0]$, where $\eta_{\min}''>0$ is the least of the numbers $\eta''=\eta'\min(1,\sqrt{I_{B}D_{u}},D_{u})$ of Corollary 8.10 over the finitely many rows, and $|\operatorname{Re}G(-\sigma+iY)|\leq d_{1}$ for every Gram test and every $\sigma=s_{1}-\delta-\delta'$ that occurs (Lemma 8.6). Then the buffer $C_{B}>\max(C_{P}+M,1.5)$ with $4c_{\mathrm{out}}/(H_{0}C_{B}^{2})\leq\eta$.

(S5) The tolerances of the published results and of the lemmas proved here, each below the finitely many positive margins it must fit: the tolerances of Inputs 3.3 and 3.4 for the finitely many test functions (those of Section 5, the functions $f_{\sigma}^{\theta}$ with tolerance $H_{0}\varepsilon_{1}/8$, and the detectors of the near rows with tolerance $\eta_{\min}''/2$), the disc radius $\delta_{0}\leq\frac{1}{3}$ of Section 5.3, which is at most the radius that Input 3.4 provides for each of these, and the tolerances of (12) and Lemma 7.8, of Input 6.1 (with $\varepsilon=\eta J^{2}$), of Inputs 3.7 and 3.8 and Mertens’ estimate, of Input 4.2 (with $\varepsilon=\eta H_{0}$), of the zero tables, and of the small-branch estimates of Theorem 12.5 (with $E+\varepsilon<0.03u_{\mathrm{exc}}$).

(S6) Finally $q_{0}$: the maximum of the finitely many thresholds of the previous stages, together with those of Input 3.5, of the inclusions $R_{P}\subseteq R_{B}\subseteq\text{the discs}$, and $L>4M(K_{0}+1)$ for Lemma 6.3.

The certificates of stage (S1) depend only on fixed data, not on $C_{P}$, $M$, $C_{B}$ or $q$. The count $K_{0}$ counts zeros up to height 2, which exceeds $1+\delta_{0}$ for every admissible radius; so $M$, which depends on $K_{0}$, can be fixed before the radius $\delta_{0}$, and there is no circular dependence. The strip height $T^{*}=T^{*}(q)\in[\frac{1}{2},1]$ of Lemma 6.3 is then determined by $q$.

Envelope errors that are summed over unboundedly many characters are weighted by $e^{-A\lambda_{\chi}}$, and $\sum_{\chi}e^{-A\lambda_{\chi}}<19$ by (10); errors in the near rows are uniform per entry or are multiples of $(\sum_{j}a_{j})^{2}$ (Theorem 8.4). Thus the error allowances remain uniform as the number of characters grows with $q$.

The regimes

Let $q\geq q_{0}$. Exactly one of the following holds.

(R1) No nonprincipal $L(s,\chi)$ has a zero in $R(l)$.

(R2) $\lambda_{1}<0.1$.

(R3) $0.1\leq\lambda_{1}<1.5$.

(R4) $\lambda_{1}\geq1.5$.

(The first zero $\lambda_{1}$ is defined when (R1) fails; since $R(l)$ is compact there are finitely many zeros in it.)

Proof of Theorem 1.1

Let $q \ge q_{0}$ and let $a$ be coprime to $q$.

In case (R1), $W=0$ and Proposition 4.4 gives a prime $p \equiv a \pmod{q}$ with $q^{A}<p<q^{L}$.

In case (R2), if $\lambda_{1}<u_{\mathrm{exc}}$ then $P(a,q)\le q^{3.5}$ by Proposition 12.1; if $u_{\mathrm{exc}}\le\lambda_{1}<0.1$ then Theorem 12.5 gives a prime $p\equiv a\pmod{q}$ with $q^{3.6258}<p<q^{3.99}$.

In case (R3), Theorem 11.7 gives a leaf of the case tree, or a subcase of a leaf, and a feasible point of its program with $W\le\mathcal{V}(x)-2\eta<1-2\eta$. In case (R4), Theorem 12.6 gives $W<1-2\eta$. In both cases Proposition 4.4 gives a prime $p\equiv a\pmod{q}$ with $q^{A}<p<q^{L}$.

Thus $P(a,q)<q^{3.99}$ for every $q\ge q_{0}$ and every $(a,q)=1$. There are only finitely many moduli $2\le q<q_{0}$, and finitely many reduced residue classes for each such modulus. Dirichlet’s theorem supplies a least prime in each class. Taking the maximum of 1 and the finitely many ratios $P(a,q)/q^{3.99}$ gives an absolute $C$ valid for all $q\ge2$. $\square$

Remark 13.1 (Effectivity). Heath-Brown’s theorem on Siegel zeros [50], used for the closest exceptional zeros, is effective. The other source estimates used here are also stated with effective constants. However, we have neither audited the effectivity of every step of the assembly nor computed a value of $q_{0}$ or $C$. Theorem 1.1 asserts their existence and does not supply an explicit numerical bound for either constant.

The computations and their verification

The finite computation has three logical components. The zero-location checks justify the case analysis; enclosures of analytic quantities supply the leaf coefficients; and integer certificates bound the resulting programs. The exterior ranges require separate computations. A check of one component does not by itself verify the others. This section records the evidence for each and the scope of the Lean formalization.

Computer assistance is well established in explicit analytic number theory, for instance in the ternary Goldbach problem [54] and in verifications of the Riemann hypothesis [106, 107], and linear programs with rigorously checked bounds are central to the proof of the Kepler conjecture [96, 119, 46]. In our computations, the real coefficients and margins are enclosed by interval arithmetic [88, 89, 113, 132]. It is implemented with the interval context of mpmath [126], at a working precision of at least 50 digits, and with a vectorized binary64 interval module that widens the result of every operation outward by one unit in the last place and evaluates exp, sin and cos by Taylor polynomials with explicit remainder bounds, not by the platform’s library; the Lean checkers of Section 14.3 use their own verified rational interval arithmetic. Floating-point linear programming (with the HiGHS solver [55] in SciPy [137]) is used only to propose dual multipliers, and every certificate is checked in exact integer arithmetic. Independent numerical audits use multiple-precision arithmetic [126]. The supplementary floating-point checks of certain published table entries are identified separately in Section 7.1. Table 12 summarizes, for each component of the proof, whether it is published or new, how it was obtained and how it was checked.

The data

The certificates are stored in three compressed JSON record files. The inside and outside files are indexed by root case; the supplementary file records the 233 supplementary leaves of 199 inside roots (Section 11.4). Their sizes and SHA-256 hashes are:

File	Bytes
inside.jsonl.gz	298,536,973
539362a03e7b330dad03b33d6b5037871cfcbbf7b0973e7976fccf12509fe63c
nodes.jsonl.gz	159,624,716
b715ac735b6976da17a2671667ecfdf07c3f19b8ab74ae5f0533dbcf6b2a6cfc
outside.jsonl.gz	8,626,435
a53d4d5793f71c92647a7f22022110bdc8135cbda1f2575d1dc0c1f487f5ac1d

The first holds the 2768 inside roots with their 2913 other leaves, the second the 233 supplementary leaves, and the third the 1685 outside roots with their 1642 leaves. A leaf record contains the parameters of its near rows and its certificate tree; it does not contain the leaf program itself, which the verifier regenerates from the case tree. Some leaves are split into subcases (Section 11.6), each with its own certificate tree, so the 4788 leaves carry 5591 certificate trees in all: 3949 inside (including the supplementary leaves) and 1642 outside.

Exact verification

All numbers entering a leaf program are enclosed with outward-rounded interval arithmetic and stored as integers scaled by $S=10^{16}$, rounded in the safe direction (Section 10). The verification of a certificate uses only integer arithmetic. The following checks have been run.

1. The replay. A verifier traverses the case tree from its roots. It re-checks every split, location node and exclusion (regenerating the location rows, except that the table of complex polynomial rows and the side conditions [145], eq:4.29, eq:4.34 are checked by separate programs). At every leaf it regenerates the leaf program from the specification and the tree, independently of the stored certificate, and checks the certificate exactly. It passes on all 4453 roots: 4788 leaves, 4,196,879 boxes, and largest certified value $0.9999998223\ldots<1$ (inside) and $0.9999942478\ldots$ (outside).

2. An independent checker. A second verifier was written from Section 10 alone, without reading or using the checking code of the first, and it uses only exact integer and rational arithmetic. It takes the leaf data from the same generator as the replay (the leaf data are checked separately, in (4) and in Section 14.3). It accepts every certificate of the corpus (29,397,336 relaxation cases in all) and that of Theorem 12.6, and for every certificate tree its value and number of boxes are exactly those of the first verifier. In a negative test on ten leaves, including the tightest, it rejected all 308 certificates made invalid by deliberate forgeries, among them duals lowered, and coefficients raised, by the smallest amount that breaks (10.1); it accepted all 82 controls perturbed by one unit less. It replaces an earlier independent checker, whose code was not preserved and which had also accepted every certificate of the corpus.

3. A mutation test. On the 19 leaves of three of the tightest inside roots, 133 perturbations of the regenerated leaf data (far budget, objective, first-family charge, correlation term, features), of certificate multipliers and of the allowance $5\eta$, each in the unfavourable direction, were all rejected by the replay.

4. Audits of the constants. For 38 leaves of all kinds, the 19,616 objective and 19,616 far coefficients, the tail, hidden and second-family columns, $F$, first and all 120 family, shifted and graded near rows were recomputed from their definitions in multiple-precision arithmetic by a program sharing no code with the builders; all lie on the safe side of their true values. (The program of this audit was not preserved; the numeric and row checkers of Section 14.3 now check the same quantities for every leaf, except the two-test rows.) Separately, for 117 roots (100 of them chosen at random), all 530,942 column and budget integers of their 154 leaves (objective, far, count, hidden-count and second-family

coefficients, $F$, first and final) were recomputed from their defining formulas in 50-digit arithmetic and agree exactly with the stored ones.

(5) The two-test rows. A separate program replays the 37 roots whose leaves use the row of Section 9.6 (31 with a single test, 2 with a mixture, 4 paired). It compares the row as the certificates use it with a separate implementation of the same inequality, with its own builder and its own certificate checker. The comparison runs on every threshold interval that the 37 certificates use (1079 intervals, from 23,341 boxes) and on 11,211 seeded random intervals, in the first-order relaxation and in both tangent relaxations (for these, the tangent rule applied to the exact inequality of that implementation). The exact costs and budgets are those of that inequality divided by $D$, and the integers used are rounded in the safe direction. The row’s integers are regenerated by the interval routines of that implementation and agree with the leaves; every one-unit perturbation of them is detected. These constants rest on those interval routines alone.

The exterior regimes. The margins of Theorem 12.5 were regenerated from their definitions and checked in interval arithmetic by a separate replay of the exterior components, and they have been recomputed independently in multiple-precision interval arithmetic. The program of Theorem 12.6 is a leaf program in the sense of Definition 10.1. Its objective and far integers are regenerated and compared with the stored ones, and its near row is built with the normalization of Section 9.2. The exact checking code of the replay (1) and the independent checker (2) accept its certificate, and so do the verified checkers of Section 14.3, which also accept its numeric data and its near row.

Formal verification

Several parts of the argument have been formalized in the Lean 4 proof assistant [29] with the Mathlib library [125]; see [2] for the role of such formalizations. Earlier formalizations in analytic number theory include the prime number theorem [3, 47, 72] and Dirichlet’s theorem [121]. The formalization uses Lean v4.35.0-rc3 with the corresponding Mathlib release, and contains about 24,000 lines and about 800 theorems and lemmas. The 18 statements of the near lemma and the certificate checker were checked against their proofs with the Comparator tool [75], which replays the proofs in the Lean kernel and in the independent kernel nanoda [5]; the leaf-level and near-row theorems were checked by the Lean kernel. All proofs are complete and use only the three standard axioms (propositional extensionality, the axiom of choice, and the soundness of quotients). The published inputs of Section 3 enter as explicit hypotheses, each stated no more strongly than its printed source. The following results are formally verified.

(a) The graded near-density lemma (Theorem 8.4), from the local explicit formulas, Burgess’s estimate and Graham’s estimate as hypotheses. This includes the response lemma, the threshold form, and the entry bounds of Section 9.

(b) The soundness of the certificate checker of Section 10: an accepted certificate proves that every feasible point of the leaf program has value below 1.

(c) The passage from a certified leaf with valid data to primes. From Xylouris’s criterion and his far-density lemma, stated as printed, and the localized envelope of Proposition 5.5, stated as a hypothesis, a leaf whose certificate and data pass the checks yields a prime $q^{A}<p<q^{L}$ in every reduced class, for every modulus whose zeros satisfy the leaf’s zero-level hypotheses. The count, hidden-count and second-family integers and final enter as they are (they were recomputed in the audit of (4) above).

(d) Interval arithmetic with proved inclusion, including the exponential function, and enclosures of the functions entering the objective and far data of the leaves. A numeric checker is proved to imply that each leaf’s objective and far coefficients, far budget and first-family charge are valid.

(e) Heath-Brown’s Conditions 1 and 2 for the parabolic test functions of Section 9.

(f) The near rows: each family, shifted and graded near row is a concrete instance of Theorem 8.4. A row checker is proved sound, including the three family terms that keep a second zero and the constant $C_Z$ of Section 9. A checked row is a valid row of the leaf program under zero-level hypotheses only, and this is composed with (c). (The formal version of Corollary 8.10 assumes that the numerators $D(\delta_j)-d_1+e_j$ themselves are positive, which is stronger than what the proof in Section 8 needs; the row checker verifies this for every checked entry.)

The certificate checker has a Mathlib-free copy that is proved to compute the same value; the numeric checker and the row checker are Mathlib-free themselves and are proved sound directly. Run as compiled programs, the certificate and numeric checkers accept every certificate tree and the objective, far and first-family data of every leaf in the corpus: 5591 trees and 4,196,879 boxes. The compiled row checker accepts every near row of the corpus except the 37 two-test rows of Section 9.6, which it does not cover: 16,842 rows with 11,135,323 entries. The same three checkers accept the leaf program of Theorem 12.6: its certificate (36 boxes), its objective and far data, and its near row (150 entries). So (c) and (f) apply to it as to the leaves of the case tree. In addition, seven certificates are checked by evaluation in the Lean kernel itself: six of the case tree (122 boxes) and that of Theorem 12.6 (36 boxes).

The published inputs (Sections 3, 7 and 12) enter the formal statements as hypotheses. The following parts have conventional proofs in this paper but are not formally verified. Their numerical checks are as described here and in the relevant sections; in particular, a multiple-precision audit is distinct from a formally verified enclosure.

• the per-character costs of Section 5: the localized envelope, which enters as a hypothesis, and the first-family bounds (the enclosures of $J_{\mathrm{old}}$ and $J_{\mathrm{new}}$ are verified, but not the inequalities they bound);

• the conductor coefficient $\frac{1}{4}$ for real characters and the refinement of Section 7.6;

• the zero-location arguments and the case tree (Sections 7 and 11), that is, the statement that every configuration of zeros falls into a leaf whose zero-level hypotheses it satisfies, including the count, hidden-count and second-family constraints;

• the exterior regimes (Section 12): the range of Proposition 12.1 and Theorem 12.5, and, for Theorem 12.6, the statement that every configuration with $\lambda_1 \ge1.5$ satisfies the zero-level hypotheses of its program;

• the two-test row of Section 9.6, which is checked by the replay and compared exactly with a separate implementation in (5), and whose constants were computed only by the interval routines of that implementation;

• the count-type integers of the leaf programs (checked in the audit of (4));

• the assembly of Section 13.

A compiled program is trusted to compute what it is proved to compute; its correctness then rests on the Lean compiler and runtime and on the unverified routines that read the data files. The Lean proofs were written by AI agents, like the rest of this work (see the note after the abstract). The Lean kernel checks the proofs, but whether each formal statement says what the corresponding statement of this paper says has so far been reviewed only by AI agents.

Data availability

The development repository contains the certificates, the programs that generate and check them, the verification reports, and the Lean formalization:

https://github.com/enaslund/linniks-constant.

At the date of this revision the repository is private. A public, versioned archive of these materials is needed before submission so that readers can reproduce the computations and identify the precise data used in the proof. This paper and the Lean formalization are publicly available in the repository prepared for the Palomar registry of formalized mathematics:

https://github.com/enaslund/linniks-constant-3.99.

The repository includes instructions for rerunning the replay, mutation tests, numerical audits, and formal checks. On the machine used, the replay took 1.7 hours of wall-clock time with parallel workers, and the compiled Lean checks of all certificates, leaf data and near rows took about 5 hours of wall-clock time. One caveat concerns exact reproducibility: the sieve heights of the graded rows (Section 9.2) are regenerated in binary64 floating point and are not stored in the records. Any nonnegative heights are admissible, so this does not affect soundness, but on a platform whose elementary functions round differently the regenerated rows, and hence the certificates that use them, could differ and would have to be recomputed. Thus the reproducibility requirement concerns the particular stored certificates; admissibility of the analytic sieve weight alone does not guarantee that a different regenerated row will pass those certificates.

Concluding remarks

The difficult configurations. The tightest certified cases have a nonreal first character with $\lambda_{1} \in[0.64, 0.80]$ and a nearby second family. The numerical experiments of Section 1.7 suggest that stronger location bounds in this range would be useful. One limitation of a single near row can be seen directly: if $N$ entries have the same positive feature $v$ and diagonal $D$, then its quadratic inequality permits $N \le D/(v^{2}-d)$ when $v^{2}>d$ (Example 8.9). This is a restriction of that relaxation, not a lower bound on what other tests or arguments can achieve.

Improved far-density constants may also help. Xylouris estimates [145] that halving the constant in his (5.19) would lower the exponent obtained by his method to about 4.6. A corresponding gain for the present method would require new certificates for the complete cover.

Possible improvements. Heath-Brown’s list of possible improvements [51] contains several that we have not used: optimal test functions for the combined inequalities (his item 1), positivity beyond the support of the test functions via Brun–Titchmarsh bounds (item 3; see [129, 86, 60, 81]), and modified sieve coefficients in the density argument (item 6; compare [35]). His item 7, a continuous weight in the density argument, is what Xylouris’s far-density lemma implements, and we use it. For cube-free moduli Burgess’s bounds hold for every $k$, and even the Weyl bound is known [100]; Heath-Brown already noted that all the arguments improve in that case. Recent large-value estimates for Dirichlet polynomials [44, 17, 122] and new density theorems [105] improve zero-density estimates away from $\sigma=1$; it is not clear whether they help in the range $\sigma=1-O(1/\log q)$ that matters here.

Other approaches. For moduli with bounded cubic part the exponent 4.5 has been claimed [83], and for moduli all of whose prime factors are small much better exponents are known [16]. A different recent approach, through the exceptional set of Goldbach’s problem (compare [104]), claims the exponent 5 in a preprint [148].

The role of the computation. The graded lemma gives quadratic constraints for different test functions and anchors. Their threshold forms can be combined with the far-density budget in linear programs, whose bounds are certified by exact duality. This separates the search for effective test parameters from the verification of the resulting bound. Any improvement of the global exponent must still include the zero-location arguments, a complete case cover, and the exterior regimes.

Appendix A. The complex location rows

The 94 rows below make the implications of Propositions 7.12 and 7.13 explicit. In each row, the hypothesis is that $\chi_{1}$ is nonreal and $\lambda_{1} \in[a,b]$. The number $h$ is the resulting strict lower bound for $\lambda_{2}$ or $\lambda'$, as specified in the caption. The parameters $\gamma$ and $t$ select the test function (11) and the polynomial $P_{t}$.

Each margin is the interval-certified amount, divided by $f(0)$, by which the necessary inequality fails at error tolerance 0; the displayed margins are rounded for readability. Their positivity allows a common sufficiently large $q$. For the $\lambda_{2}$ rows, the real-second-character case uses the test parameter $\gamma_{r}$ and $t'=0.85$; the column labelled “alias” gives the margin in (13). The $\lambda'$ rows use the parent lower bound described after Proposition 7.13 and assume $\operatorname{ord}\chi_{1}\ge5$. For orders 3 and 4, the stronger published bound quoted there applies.

$[a,b]$	$h$	$\gamma$	$t$	$\gamma_{r}$	generic	real	alias
$[.66,.6625]$	$.763$	$1.3$	$.95$	$1.15$	$0.00146$	$0.0323$	$0.1637$
$[.6625,.665]$	$.762$	$1.3$	$.95$	$1.15$	$0.00084$	$0.0314$	$0.1639$
$[.665,.6675]$	$.76$	$1.3$	$.9$	$1.15$	$0.00114$	$0.0311$	$0.1644$
$[.6675,.67]$	$.758$	$1.3$	$.9$	$1.15$	$0.00148$	$0.0308$	$0.1647$
$[.67,.6725]$	$.757$	$1.3$	$.9$	$1.15$	$0.00091$	$0.0299$	$0.1649$
$[.6725,.675]$	$.755$	$1.3$	$.9$	$1.15$	$0.00126$	$0.0296$	$0.1651$
$[.675,.6775]$	$.753$	$1.3$	$.9$	$1.15$	$0.00162$	$0.0293$	$0.1654$
$[.6775,.68]$	$.752$	$1.3$	$.9$	$1.15$	$0.00106$	$0.0284$	$0.1655$
$[.68,.6825]$	$.75$	$1.3$	$.9$	$1.15$	$0.00143$	$0.0281$	$0.1658$
$[.6825,.685]$	$.749$	$1.3$	$.9$	$1.15$	$0.00088$	$0.0272$	$0.1658$
$[.685,.6875]$	$.747$	$1.3$	$.9$	$1.15$	$0.00126$	$0.0269$	$0.1658$
$[.6875,.69]$	$.745$	$1.3$	$.9$	$1.15$	$0.00164$	$0.0266$	$0.1658$
$[.69,.6925]$	$.744$	$1.3$	$.9$	$1.15$	$0.00110$	$0.0258$	$0.1658$
$[.6925,.695]$	$.742$	$1.3$	$.9$	$1.15$	$0.00150$	$0.0254$	$0.1658$
$[.695,.6975]$	$.741$	$1.3$	$.9$	$1.15$	$0.00095$	$0.0246$	$0.1658$
$[.6975,.7]$	$.739$	$1.3$	$.9$	$1.1$	$0.00136$	$0.0243$	$0.1657$
$[.7,.7025]$	$.738$	$1.3$	$.9$	$1.1$	$0.00082$	$0.0236$	$0.1657$
$[.7025,.705]$	$.736$	$1.3$	$.9$	$1.1$	$0.00124$	$0.0234$	$0.1657$
$[.705,.7075]$	$.734$	$1.3$	$.9$	$1.1$	$0.00167$	$0.0232$	$0.1657$
$[.7075,.71]$	$.733$	$1.3$	$.9$	$1.1$	$0.00114$	$0.0225$	$0.1657$
$[.71,.7125]$	$.731$	$1.3$	$.9$	$1.1$	$0.00158$	$0.0223$	$0.1657$
$[.7125,.715]$	$.73$	$1.3$	$.9$	$1.1$	$0.00106$	$0.0216$	$0.1657$
$[.715,.7175]$	$.728$	$1.25$	$.95$	$1.1$	$0.00146$	$0.0214$	$0.1657$
$[.7175,.72]$	$.727$	$1.25$	$.95$	$1.1$	$0.00105$	$0.0207$	$0.1657$
$[.72,.7225]$	$.725$	$1.25$	$.95$	$1.1$	$0.00155$	$0.0206$	$0.1657$

$[a,b]$	$h$	$\gamma$	$t$	margin	$[a,b]$	$h$	$\gamma$	$t$	margin
$[.68,.6825]$	$.975$	$1.15$	$.95$	$0.00179$	$[.77,.7725]$	$.908$	$1.1$	$.95$	$0.00169$
$[.6825,.685]$	$.973$	$1.15$	$.95$	$0.00175$	$[.7725,.775]$	$.906$	$1.1$	$.95$	$0.00187$
$[.685,.6875]$	$.971$	$1.15$	$.95$	$0.00171$	$[.775,.7775]$	$.904$	$1.1$	$.95$	$0.00205$
$[.6875,.69]$	$.968$	$1.15$	$.95$	$0.00228$	$[.7775,.78]$	$.903$	$1.1$	$.95$	$0.00161$
$[.69,.6925]$	$.966$	$1.15$	$.95$	$0.00225$	$[.78,.7825]$	$.901$	$1.1$	$.95$	$0.00180$
$[.6925,.695]$	$.964$	$1.15$	$.95$	$0.00222$	$[.7825,.785]$	$.899$	$1.1$	$.95$	$0.00199$
$[.695,.6975]$	$.962$	$1.15$	$.95$	$0.00220$	$[.785,.7875]$	$.897$	$1.1$	$.95$	$0.00219$
$[.7,.7025]$	$.958$	$1.15$	$.95$	$0.00216$	$[.7875,.79]$	$.896$	$1.1$	$.95$	$0.00175$
$[.7025,.705]$	$.956$	$1.15$	$.95$	$0.00215$	$[.79,.7925]$	$.894$	$1.1$	$.95$	$0.00196$
$[.705,.7075]$	$.954$	$1.15$	$.95$	$0.00214$	$[.7925,.795]$	$.892$	$1.1$	$.95$	$0.00217$
$[.7075,.71]$	$.952$	$1.15$	$.95$	$0.00213$	$[.795,.7975]$	$.891$	$1.1$	$.95$	$0.00173$
$[.71,.7125]$	$.95$	$1.15$	$.9$	$0.00214$	$[.7975,.8]$	$.889$	$1.1$	$.95$	$0.00195$
$[.7125,.715]$	$.948$	$1.15$	$.9$	$0.00218$	$[.8,.8025]$	$.887$	$1.1$	$.95$	$0.00217$
$[.715,.7175]$	$.946$	$1.15$	$.9$	$0.00222$	$[.8025,.805]$	$.886$	$1.1$	$.95$	$0.00174$

$[a,b]$	$h$	$\gamma$	$t$	margin	$[a,b]$	$h$	$\gamma$	$t$	margin
$[.7175,.72]$	$.945$	$1.1$	$.95$	$0.00170$	$[.805,.8075]$	$.884$	$1.1$	$.9$	$0.00197$
$[.72,.7225]$	$.943$	$1.1$	$.95$	$0.00181$	$[.8075,.81]$	$.883$	$1.1$	$.9$	$0.00157$
$[.7225,.725]$	$.941$	$1.1$	$.95$	$0.00193$	$[.81,.8125]$	$.881$	$1.1$	$.9$	$0.00184$
$[.725,.7275]$	$.939$	$1.1$	$.95$	$0.00205$	$[.8125,.815]$	$.879$	$1.1$	$.9$	$0.00212$
$[.7275,.73]$	$.937$	$1.1$	$.95$	$0.00217$	$[.815,.8175]$	$.878$	$1.1$	$.9$	$0.00173$
$[.73,.7325]$	$.936$	$1.1$	$.95$	$0.00170$	$[.8175,.82]$	$.876$	$1.1$	$.9$	$0.00202$
$[.7325,.735]$	$.934$	$1.1$	$.95$	$0.00183$	$[.82,.8225]$	$.875$	$1.1$	$.9$	$0.00163$
$[.735,.7375]$	$.932$	$1.1$	$.95$	$0.00196$	$[.8225,.825]$	$.873$	$1.1$	$.9$	$0.00192$
$[.7375,.74]$	$.93$	$1.1$	$.95$	$0.00210$	$[.825,.8275]$	$.872$	$1.1$	$.9$	$0.00153$
$[.74,.7425]$	$.928$	$1.1$	$.95$	$0.00223$	$[.8275,.83]$	$.87$	$1.1$	$.9$	$0.00183$
$[.7425,.745]$	$.927$	$1.1$	$.95$	$0.00177$	$[.83,.8325]$	$.868$	$1.1$	$.9$	$0.00214$
$[.745,.7475]$	$.925$	$1.1$	$.95$	$0.00192$	$[.8325,.835]$	$.867$	$1.1$	$.9$	$0.00175$
$[.7475,.75]$	$.923$	$1.1$	$.95$	$0.00207$	$[.835,.8375]$	$.865$	$1.1$	$.9$	$0.00207$
$[.75,.7525]$	$.921$	$1.1$	$.95$	$0.00222$	$[.8375,.84]$	$.864$	$1.1$	$.9$	$0.00169$
$[.7525,.755]$	$.92$	$1.1$	$.95$	$0.00176$	$[.84,.8425]$	$.862$	$1.1$	$.9$	$0.00200$
$[.755,.7575]$	$.918$	$1.1$	$.95$	$0.00192$	$[.8425,.845]$	$.861$	$1.1$	$.9$	$0.00163$
$[.7575,.76]$	$.916$	$1.1$	$.95$	$0.00208$	$[.845,.8475]$	$.859$	$1.1$	$.9$	$0.00195$
$[.76,.7625]$	$.915$	$1.1$	$.95$	$0.00163$	$[.8475,.85]$	$.858$	$1.1$	$.9$	$0.00158$
$[.7625,.765]$	$.913$	$1.1$	$.95$	$0.00179$	$[.85,.8525]$	$.856$	$1.05$	$.95$	$0.00193$
$[.765,.7675]$	$.911$	$1.1$	$.95$	$0.00196$	$[.8525,.855]$	$.855$	$1.05$	$.95$	$0.00166$
$[.7675,.77]$	$.909$	$1.1$	$.95$	$0.00214$

Appendix B. The fixed constants

This appendix collects the constants and parameters of the proof. The error budget that they serve is summarized in Section 2.7.

References

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