Sums of Fourier coefficients involving theta series and Dirichlet characters

TL;DR

This paper derives nontrivial bounds for sums involving Fourier coefficients of cusp forms, theta series, and Dirichlet characters, extending understanding of their interactions for higher-dimensional representations.

Contribution

It provides new estimates for sums combining cusp form coefficients, theta series counts, and Dirichlet characters for  or higher, advancing analytic number theory techniques.

Findings

Established bounds for sums with -dimensional theta series and cusp form coefficients.

Extended estimates to cases with prime modulus Dirichlet characters.

Improved understanding of the interplay between Fourier coefficients and quadratic representations.

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 $[1/2,1]$.