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A SHIFTED CONVOLUTION SUM OF $d_{3}$ AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 September 2019

HENGCAI TANG
HENGCAI TANG*
Affiliation:
School of Mathematics and Statistics, Institute of Modern Mathematics, Henan University, Kaifeng, Henan 475004, PR China email [email protected]
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Abstract

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that

$$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$
This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].

Keywords

divisor functionHecke–Maass formtrace formula

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F11: Holomorphic modular forms of integral weight 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 3 , June 2020 , pp. 401 - 414
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

This project is supported by the National Natural Science Foundation of China (No. 11871193) and the Foundation of Henan University (No. CX3071A0780001).

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