Mean squares of quadratic twists of the Fourier coefficients of modular forms
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916813648953344 |
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| author | Gao, Peng Zhao, Yuetong |
| author_facet | Gao, Peng Zhao, Yuetong |
| contents | In this paper, we evaluate asymptotically a smoothed version of the sum
\[ \displaystyle \sideset{}{^*}\sum_{d \leq X} \left( \sum_{n \leq Y} λ_f(n)\Big(\frac{8d}{n}\Big)\right)^2, \]
where $\Big ( \frac{8d}{\cdot}\Big )$ is the Kronecker symbol, $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers and $λ_f(n)$ are Fourier coefficients of a given modular form $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean squares of quadratic twists of the Fourier coefficients of modular forms Gao, Peng Zhao, Yuetong Number Theory In this paper, we evaluate asymptotically a smoothed version of the sum \[ \displaystyle \sideset{}{^*}\sum_{d \leq X} \left( \sum_{n \leq Y} λ_f(n)\Big(\frac{8d}{n}\Big)\right)^2, \] where $\Big ( \frac{8d}{\cdot}\Big )$ is the Kronecker symbol, $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers and $λ_f(n)$ are Fourier coefficients of a given modular form $f$. |
| title | Mean squares of quadratic twists of the Fourier coefficients of modular forms |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.22256 |