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arXiv · 2512.05378

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

Peng Gao·Xiaosheng Wu
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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 $q$.

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Peng Gao, Xiaosheng Wu. 2025-12-05. Upper Bounds for low moments of twisted Fourier coefficients of modular forms. https://arxiv.org/abs/2512.05378

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