Moments of quadratic Dirichlet character sums

TL;DR

This paper investigates the asymptotic behavior of higher moments of quadratic Dirichlet character sums, providing new bounds and applications to $L$-functions, using advanced Tauberian techniques.

Contribution

It introduces new asymptotic formulas for moments of quadratic Dirichlet character sums and establishes bounds related to Jutila's conjecture using multivariable Tauberian theorems.

Findings

Asymptotic formulas for moments in restricted regions

Lower bounds for the exponent of the log factor in conjectures

Lower bounds for weighted averages of shifted quadratic Dirichlet $L$-functions

Abstract

We consider moments of higher powers of quadratic Dirichlet character sums. In a restricted region, we give their asymptotic behavior by using de la Bret\`{e}che's multivariable Tauberian theorem. We also give the lower bound of the exponent of $\log$ factor in the conjecture of Jutila. As an application, we give a lower bound of a weighted average of shifted moments of quadratic Dirichlet $L$-functions.