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Bounds for moments of quadratic Dirichlet character sums

Peng Gao, Liangyi Zhao

TL;DR

The paper investigates bounds for moments of smoothed quadratic Dirichlet character sums $S_m(X,Y;W)$ under the generalized Riemann hypothesis, confirming a conjecture of Jutila in this setting. The authors combine Mellin inversion and contour shifting to the critical line, together with bounds for Dirichlet $L$-functions and Hölder’s inequality, to establish upper bounds $S_m(X,Y;W) ≪ XY^m(\log X)^{m(2m+1)}$ and refined piecewise log-exponent estimates for different ranges of $m$. The results hold under GRH, with unconditional validity for $0≤m≤1$ in the un-smoothed case, and they illustrate techniques that extend to other character sums. This advances understanding of moments of quadratic character sums and supports Jutila’s conjecture in the smoothed regime, providing tools for broader applications in analytic number theory.

Abstract

We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila.

Bounds for moments of quadratic Dirichlet character sums

TL;DR

The paper investigates bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of Jutila in this setting. The authors combine Mellin inversion and contour shifting to the critical line, together with bounds for Dirichlet -functions and Hölder’s inequality, to establish upper bounds and refined piecewise log-exponent estimates for different ranges of . The results hold under GRH, with unconditional validity for in the un-smoothed case, and they illustrate techniques that extend to other character sums. This advances understanding of moments of quadratic character sums and supports Jutila’s conjecture in the smoothed regime, providing tools for broader applications in analytic number theory.

Abstract

We establish upper bounds for moments of smoothed quadratic Dirichlet character sums under the generalized Riemann hypothesis, confirming a conjecture of M. Jutila.
Paper Structure (3 sections, 2 theorems, 16 equations)

This paper contains 3 sections, 2 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['quadraticmean']}
  3. Proof of Theorem \ref{['quadraticmean1']}

Key Result

Theorem 1.1

With the notations as above and truth of GRH, we have, for large $X$, $Y$, $\varepsilon>0$ and any real $m \geq 1/2$,

Theorems & Definitions (2)

  • Theorem 1.1
  • Theorem 1.2