Shifted moments of L functions and moments of theta functions

TL;DR

This paper develops conditional upper bounds for shifted moments of Dirichlet L-functions and theta functions, extending previous work on the Riemann zeta function under the assumption of the Riemann Hypothesis.

Contribution

It introduces new conditional upper bounds for shifted moments of Dirichlet L-functions and applies these results to derive bounds for moments of theta functions.

Findings

Conditional upper bounds for shifted moments of Dirichlet L-functions

Upper bounds for moments of theta functions derived from L-function bounds

Extension of Soundararajan's and Chandee's methods to new contexts

Abstract

Assuming the Riemann Hypothesis, Soundararajan showed that $\displaystyle{\int_{0}^{T} \vert \zeta(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + \epsilon}}$ . His method was used by Chandee to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$- functions which allow us to derive upper bounds for moments of theta functions.