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Shifted moments of quadratic Dirichlet $L$-functions

Peng Gao, Liangyi Zhao

TL;DR

This work proves sharp upper bounds for shifted moments of quadratic Dirichlet L-functions under GRH, extending the Soundararajan–Harper framework to the critical-line with multiple complex shifts. The method combines a dyadic prime-decomposition, short Dirichlet-polynomial expansions, and a careful smoothing analysis to produce explicit bound factors that involve products of zeta-values at shifted arguments. A key implication is a bound for moments of quadratic Dirichlet character sums that aligns with conjectures of Jutila, via an unsmoothed analogue of smoothed bound results. The results deepen our understanding of L-value correlations on the critical line and provide tractable tools for bounding character-sum moments conditional on GRH.

Abstract

We establish sharp upper bounds for shifted moments of quadratic Dirichlet $L$-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.

Shifted moments of quadratic Dirichlet $L$-functions

TL;DR

This work proves sharp upper bounds for shifted moments of quadratic Dirichlet L-functions under GRH, extending the Soundararajan–Harper framework to the critical-line with multiple complex shifts. The method combines a dyadic prime-decomposition, short Dirichlet-polynomial expansions, and a careful smoothing analysis to produce explicit bound factors that involve products of zeta-values at shifted arguments. A key implication is a bound for moments of quadratic Dirichlet character sums that aligns with conjectures of Jutila, via an unsmoothed analogue of smoothed bound results. The results deepen our understanding of L-value correlations on the critical line and provide tractable tools for bounding character-sum moments conditional on GRH.

Abstract

We establish sharp upper bounds for shifted moments of quadratic Dirichlet -function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.

Paper Structure

This paper contains 10 sections, 12 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sums over primes
  4. Smoothed character sums
  5. Upper bounds for quadratic Dirichlet $L$-functions
  6. Proof of Theorem \ref{['t1']}
  7. Proof of Theorem \ref{['quadraticmean']}
  8. Initial treatments
  9. Proof of Lemma \ref{['Lsmooth']}
  10. Proof of Lemma \ref{['fdiff']}

Key Result

Theorem 1.1

With the notation as above and the truth of GRH, let $k\geq 1$ be a fixed integer and $a_1,\ldots, a_{k}$, $A$ fixed positive real numbers. Suppose that $X$ is a large real number and $t=(t_1,\ldots ,t_{k})$ a real $k$-tuple with $|t_j|\leq X^A$. Then where $\zeta(s)$ is the Riemann zeta function. Here the implied constant depends on $k$, $A$ and the $a_j$'s, but not on $X$ or the $t_j$'s.

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 6 more