Fractional Moments of Dirichlet $L$-Functions

TL;DR

This paper establishes bounds for fractional moments of Dirichlet L-functions at the critical line, showing that these moments grow at most polynomially with the modulus, under certain conditions.

Contribution

It proves new bounds for fractional moments of Dirichlet L-functions for specific rational exponents and extends results under the Generalized Riemann Hypothesis.

Findings

Bound $M_k(q)$ by $ o ext{const}_k imes ext{phi}(q) ( ext{log } q)^{k^2}$ for $k=1/n$

Conditional extension of bounds to all $k<2$ under GRH

Results improve understanding of L-function moments at fractional powers

Abstract

Let $k$ be a positive real number, and let $M_k(q)$ be the sum of $|L(\tfrac12,\chi)|^{2k}$ over all non-principal characters to a given modulus $q$. We prove that $M_k(q)\ll_k \phi(q)(\log q)^{k^2}$ whenever $k$ is the reciprocal $n^{-1}$ of a positive integer $n$. If one assumes the Generalized Riemann Hypothesis then the estimate holds for all positive real $k<2$.