A sharper Type I/II estimate for primes in arithmetic progressions to smooth moduli, and H_m << e^{3.8043 m}

Statement

We use the notation of Stadlmann [4], §3. Let $\varpi,\delta,\sigma> 0$. Let $\alpha,\beta$ be coefficient sequences at scales $M,N$ with $MN \asymp x$ and $N = x^\gamma$, where $\beta$ has the Siegel–Walfisz property. The moduli are the $q \le x^{1/2+2\varpi}$ dividing $P(x^\delta)$. Stadlmann proved [4 Prop. 1] that the equidistribution estimate [4 (3.28)] holds for all $\gamma\in[1/2-\sigma,1/2]$ whenever

The new condition follows from the second condition of (1) whenever $\delta\le\sigma$, and this is the case in all the applications below.

Remark 2. The theorem does not improve the exponent of distribution $1/2+1/40$ of the primes themselves [4 Thm. 1]. That deduction goes through the Heath-Brown identity and [4 Lemma 1], which require $\sigma> 1/10$. With the second condition of (1), $\sigma> 1/10$ already gives $80\varpi+ \frac{80}{3}\delta< 1$ [4 Cor. 2], so the third condition is not the binding one there. It is the binding one in the Harman-sieve application: [4 Lemma 18(II)] needs $0.40481 > 32\varpi$. OpenAI extended [4 Prop. 1], with the same three conditions, to triply densely divisible moduli [2 Prop. 2.18 and §A.4]. Their proof follows the same lemmas with the same block length, and we expect the same modification to apply. We have not checked the details and claim nothing for those moduli.

Proof of Theorem 1

All references in this section are to [4 §3], whose notation we keep: $z \in\mathbb{Z}_5(x)$, $\lambda,\tilde{\lambda} \sim\Lambda$, $w_1,w_2,m,q_0,\Delta_1,H,T(z)$ and the classes $W_6,W_7,W_8$. We write $s_2=(w_2,m)$ and $\lambda^*=(\lambda,\tilde{\lambda})$. Since $\lambda\sim\Lambda$ we have $|\lambda| \le2|\Lambda|$. We go through Lemmas 7–17 and indicate every change. Everything not mentioned is used unchanged.

Step 1: a block length depending on the pair (Lemma 10)

In the proof of Lemma 10, the variable $d$, of length $\Delta_1$, is cut by a smooth partition of unity into blocks of length $\Delta^*=\min\{N/(|\Lambda|x^{5\varepsilon}),\Delta_1\}$, which does not depend on $(\lambda,\widetilde{\lambda})$. For a fixed pair we use instead

Since $\lambda^*\ge1$ and $|\lambda|\le2|\Lambda|$, we have $\Delta^*\le2\Delta_\lambda^*$. The block length is used in three places, and each still holds with $\Delta_\lambda^*$:

(a) the number of blocks is $\ll\Delta_1/\Delta_\lambda^*$, and the cut-off functions $\tau_m$ satisfy the same derivative bounds, since $\Delta_\lambda^*\le\Delta_1$;

(b) the Taylor expansion (3.13) of $\psi_N^*((\lambda/\lambda^*)n_1+F_kd)$ in $d$ needs $|F_k(d-x_0)|/N\ll x^{-5\varepsilon}$ for $|d-x_0|\ll\Delta_\lambda^*/z_1$. Since $0<F_k\le|\lambda|/\lambda^*$,

(c) in Step 2 of the proof of Lemma 11, the expansion (3.15) needs

Since $H_kf_4\le[f_2/f^*,f_3/f^*]\le w_1$, $f^*\ge1$ and $w_1\mid z_1$, the left side is at most $(w_1/z_1)x^{-5\varepsilon}\le x^{-5\varepsilon}$.

Thus Lemma 10 holds with $\Delta_\lambda^*$ in place of $\Delta^*$, and Lemma 11 holds with $W_8$ defined by $\Delta_2\le\Delta_\lambda^*$ instead of $\Delta_2\le\Delta^*$.

Step 2: the length of the $n_1$-sum (Lemmas 11 and 14)

After the substitution $n=(\lambda/\lambda^*)n_1+F_kd$ in Lemma 10, the variable $n_1$ runs over a coefficient sequence at scale $(\lambda^*/\lambda)N$. In Lemma 11 it becomes $n_3$, at scale $(\lambda^*/(\lambda w_3))N\le N_\lambda$. The definition of $W_8$ records only $N_2\le N$; we keep $N_2\le N_\lambda$. Lemma 13 applied as in the proof of Lemma 14 then gives, for this pair,

Lemmas 15 and 16 do not involve the block length. The range of $k$, the condition $(\cdot,m)\le T(z)$ and the sequence $(\xi_k)$ are the same as in [4]. They give $\Sigma_6(z,w_6,k)\ll_\varepsilon\xi_kx^{4\varepsilon}s_2T(z)F_\lambda/q_0$ with $(\xi_k)\in\Xi(z)$. The proof of Lemma 9 is linear in the target bound: if $\Sigma_6\ll\xi_kI$ for all relevant $k$, then $\Sigma_5(z)\ll q_0(q_0,\ell)I$. Hence

Expanding $F_\lambda$ gives four terms:

By $N_\lambda\le N$ and $\Delta^*\le2\Delta_\lambda^*$, the first and the last are at most once and twice the corresponding terms $\Delta_1N/m$ and $(\Delta_1/\Delta^*)m$ of [4]. For the middle two, consider the two possible values of $\Delta_\lambda^*$. If $\Delta_\lambda^*=N_\lambda x^{-5\varepsilon}$ then $(\Delta_1/\Delta_\lambda^*)N_\lambda=x^{5\varepsilon}\Delta_1$, and otherwise $(\Delta_1/\Delta_\lambda^*)N_\lambda=N_\lambda$. Hence

Step 3: summing over the pairs (Lemma 7)

In the proof of Lemma 7, $\Upsilon_T\ll_\varepsilon\Upsilon_T+\sup_qx^\varepsilon\Upsilon_S(q)$, and the bound $\Upsilon_S(q)\ll_\varepsilon x^{-48\varepsilon}N^2$, which includes the diagonal terms, gives the term $O(x^{-47\varepsilon}N^2)$. Here $\Upsilon_T$ is a sum over pairs $(y,\widetilde y)$ of the inner sums $\Sigma_4(z_1,z_2,z_3,(y,\widetilde y))$. Stadlmann bounds it by the number of pairs, $\ll\mathcal P:=\max\{x^{\delta+10\varepsilon}H^3,H^4\}/s_2$, times the supremum. The condition $(\cdot,m)\le T$ in $\Sigma_4$ does not depend on the pair. So, by the triangle inequality, we may instead sum bounds that depend on the pair: $\Upsilon_T\le\sum_{(y,\widetilde y)}|\Sigma_4(\cdot,(y,\widetilde y))|$. The terms $\Upsilon_S(q)$ are not changed. By the proof of Lemma 8, each $\Sigma_4(\cdot,(y,\widetilde y))$ is zero or equals $\Sigma_5(z)$ for a $z\in\mathbb{Z}_5$ with $y=w_1\lambda$, $\widetilde y=w_1\widetilde\lambda$ and $\Lambda=Y/w_1$. In particular

By (3), the target $\Sigma_3\ll q_0^2(q_0,\ell)^2N^2x^{-27\varepsilon}$ of Lemma 7 (hence of Lemma 8) follows from

Condition (3.29) of [4] is the same inequality with $\sum F_\lambda$ replaced by $\mathcal{P}F$, where $F$ is the expression with $\Delta^*$ and $N$. By Step 2, the contribution of the first and last terms of $F_\lambda$ is at most twice Stadlmann’s. It is therefore covered by (3.30) and (3.32) of [4], which give the conditions $\gamma\le1/2-4\varpi-2\delta-50\varepsilon$ and $\gamma\ge1/4+12\varpi+4\delta+100\varepsilon$. It remains to replace (3.31), which concerns the two middle terms and needed $\gamma\ge32\varpi+10\delta+400\varepsilon$. By (4),

For fixed $y\ne0$, the integers $\widetilde y\sim Y$ lie in an interval of length $\lvert Y\rvert$, and $\lvert y\rvert\le2\lvert Y\rvert$. Hence

Since $\lvert y\rvert\ge\lvert Y\rvert$, it follows that $\sum_{(y,\widetilde y)}(y,\widetilde y)/\lvert y\rvert\ll x^\varepsilon\mathcal{Y}$, where $\mathcal{Y}$ is the number of admissible $y$. By the proof of Lemma 7, $\mathcal{Y}\ll\min\{H^2/s_2+H,\lvert Y\rvert/s_2+1\}$ and $\lvert Y\rvert\ll x^{\delta+5\varepsilon}H^2/(v_1,v_2)$. We also use

First contribution. We have $\max\{1,s_2/H\}\mathcal{P}\le\max\{x^{\delta+10\varepsilon}H^3,H^4\}\le x^{\delta+10\varepsilon}H^4$. Hence

This is $\ll q_0^2N^2x^{-31\varepsilon}$ provided $x^{2\delta+146\varepsilon}H^6\ll q_0^2N$, which follows from $\gamma\ge24\varpi+8\delta+190\varepsilon$.

Second contribution. We claim that $\max\{1,s_2/H\}\mathcal{Y}\ll x^{\delta+5\varepsilon}H^2$. If $s_2\le H$, then $\mathcal{Y}\le2H^2$. If $s_2>H$, we use $s_2\le2\lvert Y\rvert$: indeed $s_2\mid w_2\mid y$ for every admissible $y\ne0$, and the sum is empty otherwise. Then $(s_2/H)(\lvert Y\rvert/s_2+1)\le3\lvert Y\rvert/H\ll x^{\delta+5\varepsilon}H$. Therefore

which is $\ll q_0^2N^2x^{-31\varepsilon}$ provided $x^{3\delta+192\varepsilon}H^6\ll q_0^2N$. This follows from $\gamma\ge24\varpi+9\delta+240\varepsilon$.

Step 4: conclusion (Lemma 17 and Proposition 1)

Lemma 17 therefore holds with $\max\{1/4+12\varpi+4\delta+100\varepsilon,24\varpi+9\delta+400\varepsilon\}\le\gamma\le1/2-4\varpi-2\delta-50\varepsilon$. The final step of the proof of Proposition 1 is unchanged. It covers $\gamma$ near $1/2$ by [3 Thm. 2.8(ii), (iv)] when $68\varpi+14\delta<1$ and $72\varpi+24\delta+200\varepsilon<1$. Hence the range $\gamma\in[1/2-\sigma,1/2]$ is covered as soon as $1/4+12\varpi+4\delta<1/2-\sigma$, $24\varpi+9\delta<1/2-\sigma$ and $72\varpi+24\delta<1$, with $\varepsilon$ small enough. These are the conditions of Theorem 1. □

Consequences

We use the general Harman-sieve construction of Stadlmann [5 Def. 9 and Prop. 2], with parameters $\xi_1,\xi_2,\xi_3$. It is the construction of [4 §4], with the numerical values 0.33856, 0.40481, 0.59519, 0.19, 0.405, 0.595 replaced by $\xi_1,\xi_2,1-\xi_2,1-2\xi_3,\xi_3,1-\xi_3$. Its proof requires the five inequalities

Its input is equidistribution, on the family of moduli under consideration, for three types of convolutions: (I) smooth factors of length $\ge x^{\xi_1}$;

(II) bilinear forms with both factors Siegel–Walfisz and one length in $[x^{\xi_2},x^{1-\xi_2}]$;

(III) three smooth factors of lengths in $[x^{1-2\xi_3},x^{\xi_3}]$ with pairwise products $\ge x^{1-\xi_3}$

In [5] it is stated for the moduli $Q^*$ of that paper; its proof, like that of [4 §4] where the moduli are $x^\delta$-smooth, uses these inputs on the family of moduli concerned. In our application the coefficients of type (I) with lengths in $[x^{\xi_2},x^{1-\xi_2}]$ are treated by (II), which uses that they satisfy the Siegel–Walfisz property [2 Lemma A.8]. We include the corrections of OpenAI [2 footnotes 2–4 and Lemmas A.8–A.9], written there for fixed numerical values. The numerical inequalities in their arguments reduce to (6), $6(1-2\xi_2)>1$, $\xi_2>2/5$ and a level of distribution at most $x^{0.53}$; for instance their two grouping gaps are $1-\xi_1-3\xi_2/2$ and $4-\xi_1-9\xi_2$. The minorant $\rho$ has mass $1-c_1(\xi_2)$, where $c_1(\xi_2)=I_1+I_2$ is the explicit sum of two four-dimensional integrals in [5 Prop. 2]. Every constraint in these integrals becomes stricter when $\xi_2$ decreases, so $c_1$ is nondecreasing in $\xi_2$.

Proof. The intervals are nonempty. We have $1/4+12\varpi+4\delta<4/9-28\varpi/9-2\delta/9$ if and only if $544\varpi+152\delta<7$; the two bounds meet at $\varpi=7/544$, $\xi_2=55/136$. Also $1/4+12\varpi+4\delta<55/136$ for $\varpi<7/544$ and $\delta$ small, and $2/5<\min\{4/9-28\varpi/9-2\delta/9,55/136\}$ since $\varpi<1/70$. So the interval for $\xi_2$ is nonempty. Since $2/5<\xi_2<55/136$, we have $(1-\xi_2)/2<3/10$, $1-3\xi_2/2>0.393$ and $4-9\xi_2>49/136>0.3602$, while $1/4+7\varpi+2\delta<0.3401$; so the interval for $\xi_1$ is nonempty. The program check_parameters.py checks all the inequalities used below in exact rational arithmetic, at sample values of $(\varpi,\delta)$ (including values close to $7/544$ and small values of $\varpi$) and at points close to the ends of the intervals for $\xi_1$ and $\xi_2$. In general they follow from the displayed choices of $\xi_1,\xi_2$ by the elementary verifications below.

Type II. We use Theorem 1 with $\sigma=1/2-\xi_2+\tau$ for a small $\tau>0$ (this also absorbs the fixed slack $\epsilon$ of [5 Def. 9], which we take small depending on $\varpi$); both factors are Siegel–Walfisz, so they may be interchanged. Its conditions read $\xi_2-\tau>1/4+12\varpi+4\delta$, $\xi_2-\tau>24\varpi+9\delta$ and $72\varpi+24\delta<1$, all satisfied for $\tau$ small. If (II) instead, one would also need $\xi_2>32\varpi+10\delta$; together with the Type III bound $\xi_2<4/9-28\varpi/9$ this forces $\varpi<1/79$ (as $\delta\to0$), and Stadlmann takes $\varpi=0.01265<1/79$.

Type III. We use [3 Thm. 2.8(v)] with $\sigma_3$ strictly between $1/18+28\varpi/9+2\delta/9$ and $1/2-\xi_3$, the slack $\epsilon$ in the ranges of (III) being smaller than these gaps, exactly as in [4 Lemma 18(III)] and [2 proof of Prop. 2.22].

Type I. Smooth factors of length between $x^{\xi_1}$ and $x^{1/2}$ are covered by [2 Lemma 2.20], which needs $\xi_1>1/4+7\varpi+2\delta$ and holds for singly densely divisible moduli, for $x^\delta$-smooth squarefree moduli. The coefficients in the decomposition satisfy the Siegel–Walfisz property [2 Lemma A.8]. So lengths in $[x^{\xi_2},x^{1-\xi_2}]$ are covered by Type II, and lengths above $x^{1-\xi_2}>x^{1/2+2\varpi}$ by [2 (2.18)]. This is the argument of [2 proof of Prop. 2.22], with $(a_*,b_*,c_*)=(\xi_2,1-\xi_2,\xi_1)$.

Construction. The inequalities (6) hold. Indeed, $\xi_1+9\xi_2<4$, $2\xi_1+3\xi_2<2$ and $2\xi_1+\xi_2>1$ are imposed, $\xi_2=\xi_3$, and $17\xi_2<17\cdot55/136=6.875<7$. Moreover $6(1-2\xi_2)>1$, $\xi_2>2/5$, and the level is $1/2+2\varpi<0.53$. The three properties then follow from the proof of [5 Prop. 2] and of [4 §4], with the corrections of [2 §A.6], run on the family of $x^\delta$-smooth squarefree moduli with the inputs above. We have checked that the numerical inequalities used in these proofs reduce to those listed; we have not rewritten the whole construction. Since $\xi_2<55/136$, the monotonicity of $c_1$ gives $c_1(\xi_2)\le c_1(55/136)$. ∎

The value $c_1(55/136)$ is computed by the program deficit_c1.py, in two ways. Nested Gauss–Legendre quadrature, with each range split where its limits change formula, gives $6.604 \cdot10^{-6}$. A Monte Carlo integration over the regions written directly from the summation conditions gives $(6.628 \pm0.020)\cdot10^{-6}$. Thus $c_1(55/136) < 6.7\cdot10^{-6}$; this is a numerical, not a certified, value. As a check, the same program gives $c_1(0.40481) \approx9.33\cdot10^{-6}$, in agreement with Stadlmann’s published bound $c_1(0.40481) \le2\cdot10^{-5}$ [4 Prop. 2].

For comparison, the minorant of [4 Prop. 2] and [2 Prop. 2.22] has $\varpi= 253/20000 = 0.01265$. Here $7/544 = 0.0128676\ldots$, and the exponent of distribution tends to $1/2 + 7/272 = 0.525735\ldots$.

Proof. By the lemma of Baker and Irving [1 Lemma 1], [4 Lemma 2], a minorant with the properties of Corollary 3 gives $H_m \ll_\epsilon\exp(2(1+\epsilon)m/(\theta(1-c_1)))$ with $\theta= 1/2 + 2\varpi$, that is, the exponent $(1+\epsilon)/((1/4+\varpi)(1-c_1))$. We let $\varpi\to7/544$ and use $c_1(\xi_2) \le c_1(55/136)$, which holds because $\xi_2 < 55/136$. Since $1/(1/4 + 7/544) = 544/143 = 3.8041958\ldots$, the exponent can be taken arbitrarily close to $544/(143(1-c_1(55/136)))$. With $c_1(55/136) < 6.7\cdot10^{-6}$ this is $< 3.804222$. Since $\xi_2 < 55/136 < 0.40481$, monotonicity also gives $c_1(\xi_2) \le c_1(0.40481) \le2\cdot10^{-5}$, and then the bound is $544/(143(1 - 2\cdot10^{-5})) < 3.804272$. □

Files

anc/check-parameters.py checks the inequalities of Corollary 3 and the two constants of Corollary 4 in exact arithmetic. anc/deficit_c1.py computes $c_1$. The outputs are in the corresponding .log files.

Acknowledgement

Technical assistance: Claude (Anthropic).

References

ORCID 0009-0009-1526-1178