Primitive roots for every admissible integer base

Introduction

Let $a$ be an integer and let $p$ be a prime not dividing $a$. The multiplicative order $\operatorname{ord}_p(a)$ is the least positive integer $k$ such that $a^k \equiv1 \pmod{p}$. The residue class of $a$ is a primitive root modulo $p$ if $\operatorname{ord}_p(a)=p-1$. Call $a$ admissible if $a\ne-1$ and $a$ is not the square of an integer. These are necessary conditions for primitive-root behavior at infinitely many primes: the order of $-1$ is at most $2$, and a nonzero square modulo an odd prime has order dividing $(p-1)/2$.

Artin’s primitive root conjecture, formulated in 1927, asserts that every admissible integer is a primitive root modulo infinitely many primes; its quantitative form predicts a positive-density asymptotic [1], Section 17.2. We prove the following lower bound.

Theorem 1.1. For every admissible integer $a$, there are constants $c_a>0$ and $x_a\ge2$ such that, for every real $x\ge x_a$,

The predicted counting scale is $x/\log x$; the theorem gives $x/(\log x)^2$ in every sufficiently large dyadic interval. In particular, it proves the infinitude assertion for each fixed admissible integer, including the prescribed base 2. The constants and the threshold may depend on the base.

The analytic part of the proof establishes a separate theorem. A finite-order Hecke character of a number field $F$ means a continuous finite-image character of the idele class group $F^\times\backslash\mathbb{A}_F^\times$. Write $\mu_m$ for the group of $m$th roots of unity.

Theorem 1.2. Let $F$ be a cyclotomic number field containing $\mu_{12}$. For every finite-order Hecke character $\eta$ of $F$, the meromorphic continuation of $L_F(s,\eta)$ has no zeros in

The principal character is included, with its pole at $s = 1$ permitted. The width is common to all fields and characters in the statement, with no conductor or height cutoff. In the proof, $F$ and $\eta$ are fixed before the auxiliary data are chosen. Constants and the points from which estimates hold may depend on them; the numerical saving in the final Mellin argument does not. This distinction is what allows a uniform splitting estimate for a growing family of fields.

Historical context and the two obstructions

Hooley proved Artin’s predicted asymptotic under the Riemann hypothesis for the Dedekind zeta functions of the Kummer fields $\mathbb{Q}(\mu_n,a^{1/n})$, with $n$ squarefree [13]. These fields encode the obstructions to maximal order. For a prime $q$ and $p \nmid aq$, the conditions

say that $p$ splits completely in $\mathbb{Q}(\mu_q,a^{1/q})$. They hold precisely when $q$ divides the index $(p-1)/\operatorname{ord}_p(a)$. Hooley’s method counts and sums such splitting conditions. Here a restricted factorization of $p-1$ confines the possible index divisors to ranges where they can all be removed.

Classical unconditional results allow a finite choice of bases. Gupta and Murty constructed a set of thirteen integers from three distinct primes, at least one of which is a primitive root modulo infinitely many primes [9]. Heath-Brown proved that at most two prime bases fail the infinitude assertion, so at least one of $2,3,5$ satisfies it [11]. Averaging over the base gives another approach. Goldfeld and Stephens established early average theorems [7, 26]; Klurman, Shparlinski, and Teräväinen obtained the expected asymptotic for almost all integers in a range as short as $1 \le a \le\exp((\log\log x)^2)$ [17]. The exceptional set may depend on $x$, so these average results do not by themselves settle a prescribed base.

Two difficulties must be addressed for a fixed base. One is analytic: the splitting fields vary with the possible index divisor. A standard simultaneous zero-free region for Hecke characters associated with an abelian extension has logarithmic width depending on the field, conductors, and height, and permits at most one simple real exceptional zero [28] (Theorem 3.1). The common width in Theorem 1.2 instead gives a common power saving. The other difficulty is the prime construction: we need many primes for which the possible index divisors fall in the range controlled by that saving. We now describe the two parts of the proof and the points where they meet.

The Hecke zero-free argument

Fix a cyclotomic field $F$ containing $\mu_{12}$ and a finite-order Hecke character $\eta$. The analytic construction uses cubic theta coefficients under sextic Kummer twists. Patterson’s cubic theta series over $\mathbb{Q}(\sqrt{-3})$ is the model for the Gauss coefficients and their transformations [24, 25]; the comparison formulas are recorded by Dunn and Radziwiłł [5] (Section 5.1 and Appendix A). The companion manuscript The Quasi-Riemann Hypothesis develops a coefficient reflection and an additive probe over that field while retaining unrestricted cube factors [23] (Part I, Sections 5–7). Its zero-free theorem treats Dirichlet $L$-functions and finite-order Hecke $L$-functions over $\mathbb{Q}\left(\sqrt{-3}\right)$. We construct the required mechanisms over each fixed $F$, using the cubic theta representation of Kazhdan and Patterson and a normalized global Whittaker factorization consistent with their correction [15, 16].

Two local calculations govern the comparison. We choose generators for which sextic reciprocity has no supplementary factor. In a reflected interaction between a prime occurring to valuation one in the varying twist and a varying squarefree coefficient prime, the sextic exponents are $1$ and $-4$. Their sum is $3$ modulo $6$, so the interaction is quadratic. The quadratic large sieve of Heath-Brown, in its number-field form due to Goldmakher and Louvel, then controls the reflected moment [12, 8].

On the Poisson side, a relative-norm formula identifies the cube of the cubic Gauss phase with the negative of a unitary Hecke-character value. A finite-order correction gives the target character $\eta$. At a prime ideal $\mathfrak p$ outside a fixed excluded set, the first squarefree theta term at the principal frequency is $-\eta(\mathfrak p)(\mathrm{N}\mathfrak p)^{-\xi}$, where $\xi$ is the Mellin variable for the coefficient norm. Together with the constant term, it supplies the local factor of the reciprocal target $L$-function. The remaining local terms give a holomorphic, nonvanishing correction in the region used at the end of the proof.

Reflection and Poisson summation thus estimate the same additive average in two ways. A conductor large sieve and zero detection control the nonprincipal frequencies. The resulting power saving for the principal inverse Mellin integral continues its Mellin transform across the line $\operatorname{Re}s=1$ and excludes zeros of the target function. The use of ideal norms keeps the numerical exponents in this argument independent of the degree of $F$. Sections 3–8 carry out this proof; Appendix A provides the theta normalization.

Controlled predecessors in a fixed progression

The construction produces many primes of the form

where every prime divisor of $r$ lies between $\exp((\log x)^{0.1})$ and $\exp((\log x)^{0.3})$. We also prescribe a residue class on which the Legendre symbol $(a/p)$ equals $-1$. This excludes $2$ as an index divisor. A nontrivial remaining index has a prime divisor in $rQ$: the factor $Q$ leads to a small elementary exceptional set, and the factors of $r$ fall within the uniform splitting estimate.

The construction starts with weighted integers $d=crQ+1$ of this shape. A sieve removes small prime factors of $d$. For a remaining composite $d=mn$, with $n$ its least prime factor, a marked bilinear estimate replaces the restriction on the cofactor $m$ by an explicit density and a weaker condition on its small prime factors. Buchstab’s least-prime-factor identity leaves a positive margin between the initial sifted mass and the mass removed in this comparison. A separate upper bound makes the contribution from two prime factors close to $\sqrt{x}$ smaller than that margin.

The marked prime estimate is an input from The Poisson–Dirichlet law for the prime factors of $p-1$ [21], Theorem 3.1. We extract its scalar coefficient criterion and verify the full character and frequency estimates for the rough-factor comparison. The weights require $d-1$ to have a prime divisor in each of several prescribed ranges; these choices are the marks. A bounded multiplier in the bilinear estimate may depend on the quotient left after every power of these primes has been removed. A congruence condition on $d=mn$ need not have this invariance. We therefore expand $mn\equiv u\pmod M$ in Dirichlet characters and separate it into factors $\lambda(m)\lambda(n)$ in the two scalar coefficient sequences. Fixed twists preserve the required long character-sum bound.

The resulting fixed-progression construction is stated separately in Proposition 2.1. It applies to any fixed data $M,c,u$ satisfying its local coprimality conditions, independently of the primitive-root application. Its proof uses Bombieri–Vinogradov distribution, a Brun–Hooley block sieve, and the continuous rough-number density [3, 29, 6, 4]. The precise elementary sieve and density forms are adapted from [20], Lemmas 2.9 and 7.4, with proofs included here.

Organization and notation

Section 2 states the controlled-predecessor and splitting estimates and combines them to prove Theorem 1.1. Sections 3–8 prove Theorem 1.2 for a fixed arbitrary $F$ and $\eta$, independently of the integer base. The analytic scale parameters in those sections are local to that proof. Section 9 returns to the fixed integer $a$ and derives the uniform splitting bound. Section 10 proves the marked coefficient and rough-factor estimates; Section 11 supplies the elementary sieve facts; Section 12 proves the fixed-progression construction. Appendix A verifies the selected Kazhdan–Patterson specialization and Whittaker normalization.

Throughout, $\log$ is the natural logarithm, and $L=\log x$ in the prime-counting arguments. The symbols $\ll$ and $O$ have their usual meaning; subscripts record permitted dependence when it matters. Auxiliary parameters are fixed before the relevant asymptotic variable tends to infinity. Uniformity in those parameters is stated explicitly. The general representation-theoretic, class-field, Hecke-continuation, and classical sieve theorems are cited at their points of use. The marked Type II argument also uses the companion papers cited in the bibliography.

Reduction to controlled predecessors and complete splitting

For a prime $p\nmid a$, failure of $a$ to generate $\mathbb{F}_p^\times$ means that some prime divides the index

We construct primes for which the possible divisors of this index belong to two ranges. A large divisor forces $p$ to divide one of a short list of integers $a^j-1$; the remaining divisors are controlled by complete splitting in Kummer fields. This is the index-divisor reduction underlying Hooley’s work [13], Sections 4–7.

The first proposition supplies the primes, with a fixed congruence that will force quadratic nonresiduosity.

Proposition 2.1 (Primes with controlled predecessors). Let $M$ be a fixed positive multiple of 8, let $c\in\{2,4\}$, and let $u$ be an integer satisfying

For every sufficiently large real $x$, there are $\gg_{M,c,u} x/L^2$ distinct primes $p\in(x,2x)$ satisfying

where $r$ is a positive integer all of whose prime divisors lie in

We prove Proposition 2.1 in Section 12. For the second input, write $\mu_m$ for the group of $m$th roots of unity. For a nonzero integer $a$ and a prime $q$, put

The choice of the root $a^{1/q}$ does not affect $K_q$.

Proposition 2.2 (Uniform bound for complete splitting). Fix an integer $a$ with $|a|>1$, and put $\delta_0=10^{-6}$. For all sufficiently large real $x$ and all primes $q$ sufficiently large in terms of $a$, with $q\leq\exp(L^{0.3})$, one has

The implied constant and the threshold for $x$ are independent of $q$. For $a=2$, every prime $q\geq5$ is allowed, with absolute constants and an absolute threshold for $x$.

Section 9 proves this proposition from the Hecke zero-free theorem, Theorem 1.2. To apply it, we first record the elementary interpretation of complete splitting.

Lemma 2.3 (Splitting test). Let $a\neq0$ be an integer, and let $p,q$ be primes with $p\nmid aq$. Then $p$ splits completely in $K_q$ if and only if

Proof. The field $K_q$ is the splitting field of $T^q-a$. Fix a prime $\mathfrak{P}$ of $K_q$ above $p$. The roots are integral and have distinct reductions modulo $\mathfrak{P}$, since $p$ does not divide the polynomial discriminant $\pm q^q a^{q-1}$. Every inertia element permutes the roots and fixes their reductions, so it fixes every root. Thus $p$ is unramified. Let $\sigma$ be its arithmetic Frobenius at $\mathfrak{P}$. On $\mu_q$, this automorphism acts by $\zeta\mapsto\zeta^p$, so it fixes $E_q$ exactly when $p\equiv1\pmod q$. Under this condition, with $\alpha=a^{1/q}$, we have

The division is valid because $p\nmid a$. Reduction on $\mu_q$ is injective at $p\neq q$, so the second congruence in (2.5) is equivalent to $\sigma(\alpha)=\alpha$. Together the two conditions say that $\sigma$ fixes the generators of $K_q$. Since $K_q/\mathbb{Q}$ is Galois, this is precisely complete splitting. □

We now choose a progression of quadratic nonresidues satisfying all the hypotheses of Proposition 2.1.

Lemma 2.4 (A progression of quadratic nonresidues). Let $a\in\mathbb{Z}$ be neither $-1$ nor an integer square, and set