The Pólya–Vinogradov inequality with log log q for every character

Note. The results of this paper were first made public on 15 October 2026, as release v1.0.0 of the repository https://github.com/creelie/Breakthrough-problems-3 (concept DOI 10.5281/zenodo.23383715); later versions are archived on Zenodo as 10.5281/zenodo.23563651, 10.5281/zenodo.23629139, 10.5281/zenodo.23629649, 10.5281/zenodo.23637263 and 10.5281/zenodo.23638793. An earlier and incomplete version of the paper appeared on Hexagon on 15 October 2026 as the preprint hexagon:2610.00256, available at https://hexagonmath.org/2610.00256, and was revised there on 21 October 2026. That version claimed the bound $\sqrt q\log\log q$ for every primitive character; it deduced it from a zero-free half-plane for all Dirichlet $L$-functions announced in an unrefereed preprint of another group, which was cited as an external input, and it had a short abstract. The second version, release v2.0.0 of the same repository, archived as 10.5281/zenodo.23671158, proved every statement except that one in the paper, the zero-density theorem of Montgomery included, from Chebyshev’s bounds alone, and stated the bound for every character as a theorem resting on the zero-free half-plane $\operatorname{Re} s > 7/8$ of that preprint, whose machine-checked proof the authors compiled and checked. In this version the paper is complete in itself: Appendix B reproduces in full, with credit to OpenAI, their proof of the zero-free half-plane $\operatorname{Re} s > 11/12$ for all finite-order Hecke $L$-functions over $\mathbb{Q}(\sqrt{-3})$, from which Theorem 1.5 follows by the transfer theorem, and the title states the result for every character. The source of this revision is in release v3.0.0 of the same repository, archived as 10.5281/zenodo.23672710, with programs that check the explicit constants of the proofs, compute the sums of this paper for small moduli, and contain the compiled Lean 4 proof of the half-plane $\operatorname{Re} s > 7/8$ as an independent check.

Introduction

Let $\chi$ be a primitive Dirichlet character modulo $q \ge3$ and put

The Pólya–Vinogradov inequality states that $M(\chi) \ll\sqrt{q}\log q$ [6 Chapter 23].¹ Paley proved that the factor $\log q$ cannot be replaced by anything smaller than $\log\log q$: for infinitely many primitive quadratic characters $M(\chi) > c\sqrt{q}\log\log q$ [25]. Montgomery and Vaughan proved the matching upper bound $M(\chi) \ll\sqrt{q}\log\log q$ under the generalised Riemann hypothesis [22]. Without hypothesis, Granville and Soundararajan lowered the exponent of $\log q$ for characters of fixed odd order [10],² and Bober, Goldmakher, Granville and Koukoulopoulos determined the distribution of $M(\chi)/(\sqrt{q}\log\log q)$ over the characters to a prime modulus in the range where that ratio is at most $\log\log q$ minus a large constant [4].

Montgomery and Vaughan use the Riemann hypothesis in one place: on the major arcs of a circle method, for the prime sums of the characters $\chi\psi$ twisted by additive characters, where $\psi$ runs over the characters of a small modulus $r$. What that step needs is a power saving in these prime sums up to the height $q$, and a power saving follows from the absence of zeros of $L(s,\chi\psi)$ in a rectangle of fixed width to the left of the line $\operatorname{Re} s = 1$ and of height a power of $q$. Our first theorem makes this precise.

The number of zeros of all the $L$-functions to a modulus in such a rectangle is bounded by a zero-density theorem of the kind first proved by Ingham and Montgomery, Theorem 2.3 below, which we prove in Appendix A: the bound is a power of the modulus whose exponent tends to zero as $\theta$ tends to one, times a power of the logarithm. Counting the characters $\chi$ for which (1.1) fails gives our main result.

For a prime $q$ there are $q-2$ primitive characters, so Theorem 1.2 gives the bound of Montgomery and Vaughan for all but a vanishing proportion $q^{\varepsilon-1}$ of them. We do not know a reference for this statement. The results of Granville and Soundararajan and of Bober, Goldmakher, Granville and Koukoulopoulos on the distribution of large character sums [10, 4] are proved by a different method, through the structure of the characters with large sums, and they bound the number of characters with $M(\chi)$ above a given multiple of $\sqrt{q}\log\log q$ only in the range of multiples below $\log\log q$ minus a large constant; the complementary range, which is the one of Theorem 1.2, is where the zero-density theorem enters.

The transfer theorem also shows what a zero-free half-plane gives.

Such a half-plane exists, with $\theta= 11/12$: it is Theorem B.1 of Appendix B, due to OpenAI, and Theorem 1.5 below records what it gives.

The same zero-free rectangle controls the logarithmic derivative of $L(s,\chi)$ at $s=1$, and here the rectangle needed is small: its height is a power of $\log q$.

Theorems 1.1 to 1.4 are proved in this paper from Chebyshev’s bounds alone; the zero-density theorem they use is proved in Appendix A. The bound for every character needs a zero-free half-plane for all Dirichlet $L$-functions, and one is now known. OpenAI has proved that every finite-order Hecke $L$-function over $\mathbb{Q}(\sqrt{-3})$ has no zero with $\operatorname{Re}s > 11/12$ [24]; by base change, $L_K(s,\chi\circ N)=L(s,\chi)L(s,\chi\chi_{-3})$ for a Dirichlet character $\chi$, so no Dirichlet $L$-function has a zero there either. The proof works with the Möbius function of the Eisenstein integers: a zero to the right of the line $\operatorname{Re}s = 11/12$ would prevent a power saving in the smoothed Möbius sums of the field, and that power saving is obtained from a mean square over a family of such sums twisted by sextic residue symbols, which is estimated by two applications of Poisson summation, the transformation of Kubota’s cubic theta function at the cusps, Patterson’s formula for its Fourier coefficients and the quadratic large sieve over number fields. Appendix B reproduces that proof in full, with credit to its authors, so that the paper is complete in itself. The same group had earlier proved the weaker half-plane $\operatorname{Re}s > 7/8$ with a proof written in Lean 4 and checked by the Lean kernel against the Mathlib library [23]; we compiled that proof from its source and ran the kernel check ourselves, and the build, its axiom report and a Lean proof of the two bounds below from that input are in the archive cited in the data availability section, as an independent check of the conclusion.

Proof. By Theorem B.1, $L(s,\xi) \ne0$ for $\operatorname{Re}s > 11/12$ for every nonprincipal Dirichlet character $\xi$. The first bound is then Corollary 1.3 with $\theta= 11/12$. For the second, $L(s,\chi) \ne0$ in the rectangle of Proposition 7.1, which gives the bound. $\square$

The constants of Theorem 1.5 are absolute. Since the half-plane $\operatorname{Re}s > 7/8$ of [23] is wider, the same proof with $\theta= 7/8$ gives the two bounds with the smaller constants $C_{7/8}$ and $C_5$; the statement with $\theta= 11/12$ is the one whose proof is in this paper. Under the generalised Riemann hypothesis the bound $\left|L'/L(1,\chi)\right| \ll\log\log q$ holds for every nonprincipal character [18]. The constants $C_\varepsilon$ of Theorems 1.2 and 1.4 tend to infinity as $\varepsilon$ tends to zero, since $\theta= 1 - \varepsilon/48$ enters the proof through the width $1 - \theta$ of the zero-free rectangle; in the proof they grow like a power of $1/\varepsilon$. The exceptional characters are those whose $L$-function, or one of whose twists by a character of small modulus, has a zero near the line $\operatorname{Re}s = 1$ at a height at most $5q^2$; the theorems say nothing about those characters, and the Euler–Kronecker constant of $\mathbb{Q}(\zeta_q)$, which is a sum over all the characters modulo $q$, is not controlled by them.

The proof

Figure 1 shows how the parts fit together. Lemma 5.1 reduces $M(\chi)$ to the sums $\sum_{m\le H}\overline{\chi}(m)e(m\beta)/m$ with $H\le q/2$. On the major arcs, where $\beta$ is close to a fraction with denominator at most $R$, the integers $m$ with a large prime factor are written as $pm'$, the additive character $e(pmb/r)$ is expanded in the multiplicative characters $\psi$ modulo $r$, and the prime sums of $\chi\psi$ are bounded with a power saving by a contour shift for the Riesz mean $\sum_{n\le x}\chi\psi(n)\Lambda(n)(1-n/x)$, Proposition 3.7. The contour is a rectangle of height $x^2$ and not a whole vertical strip, which is why the hypothesis (1.1) is a rectangle and not a half-plane: on the line $\operatorname{Re}s=2$ the kernel $x^s/(s(s+1))$ decays like $|\operatorname{Im}s|^{-2}$, so the part of the integral above the height $x^2$ is bounded by a constant. On the minor arcs the sum is bounded by Theorem 4.1, a weak form of the theorem of Montgomery and Vaughan on exponential sums with multiplicative coefficients, whose proof needs only the spacing of the multiples of $\beta$ and Selberg’s upper bound for prime pairs. The integers with only small prime factors are few enough to be counted trivially. Section 6 counts the characters for which (1.1) fails: a zero of $L(s,\chi\psi)$ is a zero of the $L$-function of a character modulo $qr$, the map $\chi\mapsto\chi\psi$ is injective for fixed $\psi$, and Theorem 2.3 bounds the zeros of all the characters modulo $qr$ with $r\le R$ together. Section 7 proves Theorem 1.4 by the same contour shift, now crossing the pole of $L'/L(s+1,\chi)/(s(s+1))$ at $s=0$, whose residue is $L'/L(1,\chi)$. Appendix B proves the zero-free half-plane that Theorem 1.5 feeds into the transfer theorem; Figure 4 there shows its parts. The counting arguments are of the same kind as the counts attached to finite complexes and group actions in the authors’ earlier work [1, 2, 3].

The paper is organised as follows. Section 2 collects Chebyshev’s bounds, Selberg’s sieve and the zero-density theorem. Section 3 bounds Riesz means of prime sums by a contour shift in a rectangle. Section 4 proves the bound for exponential sums with multiplicative coefficients. Section 5 proves Theorem 1.1, Section 6 proves Theorem 1.2 and Corollary 1.3, and Section 7 proves Theorem 1.4.

Preliminaries