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A SHIFTED CONVOLUTION SUM OF $d_{3}$ AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II.
- Source :
-
Bulletin of the Australian Mathematical Society . Jun2020, Vol. 101 Issue 3, p401-414. 14p. - Publication Year :
- 2020
-
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]. [ABSTRACT FROM AUTHOR]
- Subjects :
- *TRACE formulas
*MATHEMATICAL convolutions
*DIVISOR theory
*MATHEMATICS
Subjects
Details
- Language :
- English
- ISSN :
- 00049727
- Volume :
- 101
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Bulletin of the Australian Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 143005096
- Full Text :
- https://doi.org/10.1017/S000497271900100X