The Pólya–Vinogradov inequality with $\log\log q$

Introduction

Let $\chi$ be a primitive Dirichlet character modulo $q \ge3$. The Pólya–Vinogradov inequality states that $\left|\sum_{n\le x}\chi(n)\right| \ll\sqrt{q}\log q$ uniformly in $x$ [3 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 the maximum over $x$ exceeds $c\sqrt{q}\log\log q$ [12]. Montgomery and Vaughan [9] proved the upper bound $\sqrt{q}\log\log q$ under the generalised Riemann hypothesis. Without hypothesis, Granville and Soundararajan lowered the exponent of log $q$ for characters of fixed odd order [5].² We prove the bound of Montgomery and Vaughan for every primitive character.

By Paley’s theorem, Theorem 1.1 is optimal up to the value of $C$. The input that replaces the Riemann hypothesis is the following theorem.

Theorem 1.2 is proved, and formalised in Lean 4, in [11]. It is the only result we use without proof; everything else is proved below from first principles, and the constants are effective.

Montgomery and Vaughan use the Riemann hypothesis in one place: on the major arcs, for prime sums of the characters $\chi\psi$ twisted by additive characters, where $\psi$ runs over the characters of a small modulus $r$. A power saving in these sums is enough there, and Theorem 1.2 provides one (Corollary 3.8). On the minor arcs they use their unconditional bound for exponential sums with multiplicative coefficients. A weaker form of it suffices, and we prove that form in § 4 with Selberg’s sieve as the only arithmetic input. Figure 1 shows how the parts fit together.

The same half-plane controls the logarithmic derivative of $L(s,\chi)$ at $s = 1$. For a number field $K$ write $\zeta_K(s) = c_{-1}(s - 1)^{-1} + c_0 + O(s - 1)$ near $s = 1$; the Euler–Kronecker constant of $K$ is $\gamma_K = c_0/c_{-1}$. Let $\gamma_q$ be the Euler–Kronecker constant of $\mathbb{Q}(\zeta_q)$. For a prime $q$ the factorisation $\zeta_{\mathbb{Q}(\zeta_q)}(s) = \zeta(s)\prod_{\chi\ne\chi_0} L(s,\chi)$, the product over the nonprincipal characters modulo $q$ [13 Theorem 4.3],³ gives

where $\gamma$ is Euler’s constant.

(i) For every nonprincipal character $\chi$ modulo $q$,

(ii) For every prime $q \ge3$, $-C\log q\log\log q \le\gamma_q \le C\log q$.

Under the generalised Riemann hypothesis the bound (i) is classical, Ihara proved $\gamma_q \le(2+o(1))\log q$ and Badzyan proved a lower bound of the order of (ii) [7] and [1]. Ford, Luca and Moree proved the two bounds of (ii) without hypothesis for a set of primes $q$ of density one [4] and [10].⁴ For all primes the best unconditional bound we know is $|\gamma_q| \ll_\varepsilon q^\varepsilon$, due to Ihara, Murty and Shimura [8]. The upper bound in (ii) needs only Chebyshev’s bound. The lower bound needs an upper bound for the prime powers $n \le q^{17}$ with $n \equiv1 \pmod q$, which we obtain from Selberg’s sieve applied in dyadic blocks (Lemma 6.3).

Theorems 1.1 and 1.3 are not among the consequences of Theorem 1.2 drawn in [11]. The arguments work, with other constants, whenever $L(s,\chi) \ne0$ on $\operatorname{Re}s > \theta$ for all nonprincipal $\chi$ and some fixed $\theta< 1$. The paper is organised as follows. § 2 collects Chebyshev’s bounds and Selberg’s sieve. § 3 bounds Riesz means of $\chi\Lambda$ by a contour shift. § 4 proves the bound for exponential sums with multiplicative coefficients, § 5 proves Theorem 1.1 and § 6 proves Theorem 1.3.

Preliminaries

Throughout, $p$ and $\ell$ denote primes, $\Lambda$ is the von Mangoldt function, $\theta(x) = \sum_{p\le x}\log p$, $\psi(x) = \sum_{n\le x}\Lambda(n)$, $e(x) = e^{2\pi ix}$, and $\lVert x\rVert$ is the distance from $x$ to the nearest integer. We write $\omega(n)$ for the number of prime factors of $n$, $\tau_3(n)$ for the number of ordered factorisations $n=abc$, and $P(n)$ for the largest prime factor of $n \ge2$, with $P(1)=1$.

Proof. (i) It suffices to treat integers $m \ge1$, and we use induction on $m$. The cases $m=1,2$ are clear, and for even $m>2$ we have $\theta(m)=\theta(m-1)$. Let $m=2n+1$. The primes in $(n+1,2n+1]$ divide $\binom{2n+1}{n}$, and $2^{2n+1} \le2^{2n+1}$ because $\binom{2n+1}{n}$ and $\binom{2n+1}{n+1}$ are equal terms of the binomial expansion of $(1+1)^{2n+1}$. Hence $\theta(2n+1)\le\theta(n+1)+n\log4\le(2n+1)\log4$.

(ii) We have $\psi(x)-\theta(x)=\sum_{k\ge2}\theta(x^{1/k})$, where only $k\le\log x/\log2$ contribute. Each term is at most $\sqrt{x}\log4$ by (i), so the sum is at most $(\log x/\log2)\sqrt{x}\log4=2\sqrt{x}\log x$. Since $\log x\le\sqrt{x}$, $\psi(x)\le(\log4+2)x\le4x$.

(iii) By Legendre’s formula and (i),

(iv) Let $A(u)=\sum_{p\le u}\frac{\log p}{p}$. Partial summation and (iii) give