Upper Bounds for low moments of twisted Fourier coefficients of modular forms
Peng Gao, Xiaosheng Wu
TL;DR
The paper establishes upper bounds for the low moments (2k-th with 0 ≤ k ≤ 1) of sums ∑_{n ≤ x} χ(n)λ(n) averaged over characters modulo a large prime q, where λ(n) are Fourier coefficients of a fixed holomorphic cusp form. Building on Harper's random multiplicative function framework, the authors develop a modular-form–adapted set of mean-value and probabilistic lemmas, including a smooth partition of unity and short Euler-product analysis, to compare Dirichlet-character sums with random Euler products. By transitioning to a random-model setting and controlling error terms via careful parameter choices, they prove an upper bound of the form (x/(1+(1−k)√log log L))^k, which yields an average bound o(√x) when x and q/x both tend to infinity with q. This extends prior results for Dirichlet characters to the setting of modular form coefficients and demonstrates stronger-than-square-root cancellation on average in this twisted setting. The methods have potential implications for understanding fluctuations of twisted modular-form coefficients in character sums and for related mean-value problems 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 $$\displaystyle \sum_{\substack{χ\bmod q}} \Big| \sum_{n\leq x} χ(n)λ(n)\Big|^{2k},$$ where $λ(n)$ denotes the Fourier coefficients of a fixed modular form. In particular, our result implies that $$\displaystyle \frac 1{q-1}\sum_{\substack{χ\bmod q}} \Big| \sum_{n\leq x} χ(n)λ(n)\Big|= o(\sqrt{x}),$$ when both $x$ and $q/x$ tend to infinity with $q$.