Upper Bounds for low moments of twisted Fourier coefficients of modular forms

TL;DR

This paper establishes upper bounds for low moments of twisted Fourier coefficients of modular forms, revealing their average behavior is smaller than the square root of x under certain conditions.

Contribution

It provides new upper bounds for the low moments of twisted Fourier coefficients, advancing understanding of their distribution for large primes.

Findings

Bounded the 2k-th moments for 0 ≤ k ≤ 1.

Showed the average of twisted sums is o(√x) as q and x grow.

Extended knowledge of Fourier coefficient behavior in analytic number theory.

Abstract

For any large prime $q$, $1 \leq x \leq q$ and any real $0 \leq k \leq 1$, we prove an upper bound for the following $2k$-th moment $$\displaystyle \sum_{\substack{\chi \bmod q}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k},$$ where $\lambda(n)$ denotes the Fourier coefficients of a fixed modular form. In particular, our result implies that $$\displaystyle \frac 1{q-1}\sum_{\substack{\chi \bmod q}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|= o(\sqrt{x}),$$ when both $x$ and $q/x$ tend to infinity with $q$.