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SHIFTED CONVOLUTION SUM OF
$d_{3}$ AND THE FOURIER COEFFICIENT OF HECKE–MAASS FORMS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 2 / October 2015
- Published online by Cambridge University Press:
- 02 June 2015, pp. 195-204
- Print publication:
- October 2015
-
- Article
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- You have access
- Export citation
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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 where the implied constant depends only on
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$
$t_{j}$ and 
${\it\varepsilon}$.
