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

TL;DR

This work studies the $2k$-th moments of Dirichlet $L$-functions at the central point for a fixed prime modulus $q$, proving unconditional sharp lower bounds for all $k\ge 0$ and sharp upper bounds for $0\le k\le 1$ on the critical line. The authors adapt Gao's framework by combining the Heap–Soundararajan lower bounds principle with the Radziwiłł–Soundararajan upper bounds principle and Selberg's twisted second moment, together with a multi-scale Dirichlet-polynomial construction to capture moment growth. They show the moments exhibit the conjectured order $\varphi^*(q)(\log q)^{k^2}$ in the $0\le k\le 1$ range, and provide the unconditional bounds needed to establish this order of magnitude. The results advance understanding of L-function moments for fixed moduli and furnish techniques potentially applicable to shifted moments and related averages in analytic number theory.

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 $0 \leq k \leq 1$.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6