SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENT OF HECKE–MAASS FORMS
2015 ◽
Vol 92
(2)
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pp. 195-204
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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 ${\it\varepsilon}$.
2019 ◽
Vol 101
(3)
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pp. 401-414
Keyword(s):
2014 ◽
Vol 11
(01)
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pp. 39-49
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1984 ◽
Vol 25
(1)
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pp. 107-119
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Keyword(s):
2019 ◽
Vol 15
(07)
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pp. 1487-1517
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Keyword(s):
1961 ◽
Vol 12
(3)
◽
pp. 133-138
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1991 ◽
Vol 11
(3)
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pp. 485-499
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