Upper bounds for moments of Dirichlet $L$-functions to a fixed modulus

TL;DR

The paper establishes sharp upper bounds for the $2k$-th moments, with $0\le k\le 2$, of central values of Dirichlet $L$-functions modulo a fixed prime $q$ in the primitive character family. The authors adapt the Radziwiłł–Soundararajan upper bounds framework and crucially rely on a precise evaluation of a twisted fourth moment, including an explicit mollification that shortens Dirichlet polynomials. They prove that $\sum^{*}_{\chi\pmod q}|L(\tfrac{1}{2},\chi)|^{2k} \ll_k \varphi^*(q)(\log q)^{k^2}$ for large prime $q$, which, together with known lower bounds, yields the correct order of magnitude $\asymp \varphi^*(q)(\log q)^{k^2}$. The work hinges on extending twisted fourth-m Moment results to general twists and on meticulous control of off-diagonal terms through mollification and short Dirichlet polynomials, aligning with the CFKRS predictions for moments of $L$-functions.

Abstract

We study the $2k$-th moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus and establish sharp upper bounds for all real $k \in [0,2]$.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Lemma 3.2
• proof
• Proposition 3.3
• Proposition 3.4