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Sums of Fourier coefficients involving theta series and Dirichlet characters (2410.12305v1)
Published 16 Oct 2024 in math.NT
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_12+\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 $[1/2,1]$.