Moments of quadratic Dirichlet character sums

TL;DR

This work analyzes moments of higher powers of quadratic Dirichlet character sums in a regime where the summation length $Y$ is smaller than the modulus, linking these moments to shifted moments of quadratic Dirichlet $L$-functions. The author uses de la Bretèche's multivariable Tauberian theorem to derive precise asymptotics for $S_k(X,Y)$ in a restricted region, showing a main term of the form $X Y^{k} Q(\log Y)$ with $\deg Q = 2k^2 - k$, and establishes a lower bound $c_2(k) \ge 2k^2 - k$ in Jutila's conjecture. A key contribution is a framework for lower bounds on weighted averages of shifted moments under the GRH, demonstrating that the averaged growth rate retains the $\log$-exponent $2k^2 - k$ up to a single extra logarithm. The paper also develops a robust arithmetical-analytic setup for multi-variable multiplicative functions, proving Euler products converge in a half-plane $\Re(s_j)>1/4$ and connecting these structural results to shifted moment problems for 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 Bretèche'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.

Theorems & Definitions (14)

• Conjecture 1.1: Jutila
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 3.1
• proof
• Theorem 4.1: B
• Theorem 4.2: B
• proof : Proof of Theorem \ref{['asymptotic for S_k(X,Y)']}