Bounds for Moments of Twisted Fourier coefficients of Modular Forms

TL;DR

This paper derives upper bounds for moments of twisted Fourier coefficients of modular forms and their associated L-functions, assuming the generalized Riemann hypothesis, advancing understanding of their size and distribution.

Contribution

It provides new upper bounds for moments of twisted modular L-functions and Fourier coefficient sums, under the generalized Riemann hypothesis, improving existing estimates.

Findings

Established bounds for shifted moments of modular L-functions.

Derived bounds for moments of sums involving Fourier coefficients.

Results depend on the generalized Riemann hypothesis.

Abstract

We establish upper bounds for shifted moments of modular $L$-functions to a fixed modulus as well as quadratic twists of modular $L$-functions under the generalized Riemann hypothesis. Our results are then used to establish bounds for moments of sums involving with Fourier coefficients of a given modular form twisted by Dirichlet characters.