Moments of real Dirichlet $L$-functions and multiple Dirichlet series

TL;DR

This paper studies the moments of real Dirichlet L-functions via multiple Dirichlet series, proving their meromorphic continuation and deriving asymptotic formulas for the first three moments, under certain hypotheses, with a simple method.

Contribution

It introduces a straightforward approach to meromorphically continue multiple Dirichlet series associated with Dirichlet L-functions without modifications, enabling direct derivation of moment asymptotics.

Findings

Proved meromorphic continuation of the series under the generalized Lindelöf hypothesis for k≥5.

Derived asymptotic formulas for the first three moments with power-saving error terms.

Identified swap terms in related problems, recovering recent results by Conrey and Rodgers.

Abstract

We consider the multiple Dirichlet series associated to the $k$th moment of real Dirichlet $L$-functions, and prove that it has a meromorphic continuation to a specific region in $\mathbb{C}^{k+1}$, which is conditional under the generalized Lindel\"of hypothesis for $k\geq 5$. As a corollary, we obtain asymptotic formulas for the first three moments with a power-saving error term, and detect the 0- and 1-swap terms in related problems for any $k$ (conditionally under the Generalized Lindel\"of Hypothesis), recovering the recent results of Conrey and Rodgers on long Dirichlet polynomials. The advantage of our method is its simplicity, since we don't need to modify the multiple Dirichlet series to obtain its meromorphic continuation. As a result, we obtain the asymptotic formulas directly in the form as they appear in the recipe predictions of Conrey, Farmer, Keating, Rubinstein and Snaith.