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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.12305 |
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| _version_ | 1866913548811108352 |
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| author | Yu, Yanxue |
| author_facet | Yu, Yanxue |
| contents | Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $λ_f(n)$ and
\bna
r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}.
\ena Let $χ$ be a primitive Dirichlet character of modulus $p$, a prime. In this paper, we are concerned with obtaining nontrivial estimates for the sum
\bna
\sum_{n\geq1}λ_f(n)r_{\ell}(n)χ(n)w\left(\frac{n}{X}\right)
\ena for any $\ell \geq 3$, where $w(x)$ be a smooth function compactly supported in $[1/2,1]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12305 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sums of Fourier coefficients involving theta series and Dirichlet characters Yu, Yanxue Number Theory Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $λ_f(n)$ and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let $χ$ be a primitive Dirichlet character of modulus $p$, a prime. In this paper, we are concerned with obtaining nontrivial estimates for the sum \bna \sum_{n\geq1}λ_f(n)r_{\ell}(n)χ(n)w\left(\frac{n}{X}\right) \ena for any $\ell \geq 3$, where $w(x)$ be a smooth function compactly supported in $[1/2,1]$. |
| title | Sums of Fourier coefficients involving theta series and Dirichlet characters |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.12305 |