Bounds for moments of modular $L$-functions to a fixed modulus

TL;DR

This paper establishes precise bounds for the moments of twisted modular L-functions at a fixed prime power modulus, advancing understanding of their size and distribution at the central point.

Contribution

It provides sharp lower bounds for all real moments and sharp upper bounds for moments up to order one, for twisted modular L-functions at a fixed modulus.

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 modular L-functions.

Abstract

We study the $2k$-th moment of the family of twisted modular $L$-functions to a fixed prime power modulus at the central values. We establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.