# Shifted convolution sums involving theta series

## TL;DR

This paper investigates bounds for shifted convolution sums involving Hecke eigenvalues of cuspidal newforms and representations of integers as sums of two squares, with uniformity in the shift parameter.

## Contribution

It provides new uniform bounds for shifted convolution sums involving theta series and automorphic forms, extending previous results to arbitrary levels and nebentypus.

## Key findings

- Established uniform bounds for shifted sums with respect to the shift h
- Extended analysis to forms of arbitrary level and nebentypus
- Improved understanding of correlations between Hecke eigenvalues and sums of two squares

## Abstract

Let $f$ be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus and denote by $\lambda_f(n)$ its $n$-th Hecke eigenvalue. Let $$ r(n)=\#\left\{(n_1,n_2)\in \mathbb{Z}^2:n_1^2+n_2^2=n\right\}. $$ In this paper, we study the shifted convolution sum $$ \mathcal{S}_h(X)=\sum_{n\leq X}\lambda_f(n+h)r(n), \qquad 1\leq h\leq X, $$ and establish uniform bounds with respect to the shift $h$ for $\mathcal{S}_h(X)$.