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Shifted convolution sums involving theta series

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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 nth Hecke eigenvalue. Let

$$\begin{aligned} r(n)=\#\left\{ (n_1,n_2)\in \mathbb {Z}^2:n_1^2+n_2^2=n\right\} . \end{aligned}$$

In this paper, we study the shifted convolution sum

$$\begin{aligned} \mathcal {S}_h(X)=\sum _{n\le X}\lambda _f(n+h)r(n), \qquad 1\le h\le X, \end{aligned}$$

and establish uniform bounds with respect to the shift h for \(\mathcal {S}_h(X)\).

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Acknowledgements

The author is very grateful to the referee for detailed comments and valuable suggestions which bring many improvements on the original draft.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University, Weihai, 264209, Shandong, China

    Qingfeng Sun

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  1. Qingfeng Sun
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Correspondence to Qingfeng Sun.

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This work was supported by the National Natural Science Foundation of China (Grant No. 11101239), the Natural Science Foundation of Shandong Province (Grant No. ZR2016AQ15) and Young Scholars Program of Shandong University, Weihai (Grant No. 2015WHWLJH04).

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Sun, Q. Shifted convolution sums involving theta series. Ramanujan J 44, 13–36 (2017). https://doi.org/10.1007/s11139-017-9900-y

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  • DOI: https://doi.org/10.1007/s11139-017-9900-y

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