Bounds for Moments of Dirichlet $L$-functions of fixed modulus on the critical line

TL;DR

This paper establishes precise bounds for the moments of Dirichlet L-functions at the critical line for fixed prime moduli, advancing understanding of their size and distribution.

Contribution

It provides sharp lower bounds for all real moments and sharp upper bounds for moments up to order one, filling gaps in the understanding of Dirichlet L-functions' moments.

Findings

Sharp lower bounds for all real moments $k \\geq 0$.

Sharp upper bounds for moments with $0 \\leq k \\leq 1$.

Results contribute to the understanding of the size and distribution of Dirichlet L-functions.

Abstract

We study the $2k$-th moment of the family of Dirichlet $L$-functions to a fixed prime modulus on the critical line and establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.