Upper Bounds for low moments of twisted Fourier coefficients of modular forms
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911304000733184 |
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| author | Gao, Peng Wu, Xiaosheng |
| author_facet | Gao, Peng Wu, Xiaosheng |
| contents | 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 $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper Bounds for low moments of twisted Fourier coefficients of modular forms Gao, Peng Wu, Xiaosheng Number Theory 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 $q$. |
| title | Upper Bounds for low moments of twisted Fourier coefficients of modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.05378 |