Shifted convolution sums involving theta series

Qingfeng Sun

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arXiv · 1705.00839

Shifted convolution sums involving theta series

Qingfeng Sun

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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 $\mathcal{S}_h(X)$.

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Qingfeng Sun. 2017-05-02. Shifted convolution sums involving theta series. https://arxiv.org/abs/1705.00839

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