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

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  02 June 2015

HENGCAI TANG
HENGCAI TANG*
Affiliation:
School of Mathematics and Information Sciences, Institute of Modern Mathematics, Henan University, Kaifeng, Henan 475004, PR China email hctang@henu.edu.cn
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Abstract

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Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$
where the implied constant depends only on $t_{j}$ and ${\it\varepsilon}$.

Keywords

divisor functionHecke–Maass formVoronoi summation 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 92 , Issue 2 , October 2015 , pp. 195 - 204
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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