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arXiv · 2410.12305

Sums of Fourier coefficients involving theta series and Dirichlet characters

Yanxue Yu
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Abstract

Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ 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 $\chi$ 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}\lambda_f(n)r_{\ell}(n)\chi(n)w\left(\frac{n}{X}\right) \ena for any $\ell \geq 3$, where $w(x)$ be a smooth function compactly supported in $[1/2,1]$.

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Yanxue Yu. 2024-10-16. Sums of Fourier coefficients involving theta series and Dirichlet characters. https://arxiv.org/abs/2410.12305

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