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A SHIFTED CONVOLUTION SUM OF
$d_{3}$ AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 26 September 2019, pp. 401-414
- Print publication:
- June 2020
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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 This corrects and improves the result of the author [‘Shifted convolution sum of
$$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$
$d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].
SHIFTED CONVOLUTION SUM OF
$d_{3}$ AND THE FOURIER COEFFICIENT OF HECKE–MAASS FORMS
- Part of
-
- 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
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- Article
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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 
${\it\varepsilon}$.
