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

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 September 2019

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

Keywords

divisor functionHecke–Maass formtrace 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 101 , Issue 3 , June 2020 , pp. 401 - 414
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

This project is supported by the National Natural Science Foundation of China (No. 11871193) and the Foundation of Henan University (No. CX3071A0780001).

References

Blomer, V. and Buttcane, J., ‘On the subconvexity problem for $L$-functions on $\text{GL}(3)$’, Ann. Sci. ENS. (to appear), arXiv:1504.02667.Google Scholar
Blomer, V., Harcos, G. and Michel, P., ‘A Burgess-like subconvex bound for twisted L-functions’, Forum Math. 19 (2007), 61–105.10.1515/FORUM.2007.003Google Scholar
Deshouillers, J.-M. and Iwaniec, H., ‘Kloosterman sums and Fourier coefficients of cusp forms’, Invent. Math. 70 (1982), 219–288.10.1007/BF01390728Google Scholar
Erdélyi, A., Magnus, W., Obergettinger, F. and Tricomi, F., Higher Transcendental Functions II (McGraw-Hill, New York, 1953).Google Scholar
Kim, H. H. and Sarnak, P., ‘Appendix 2: refined estimates towards the Ramanujan and Selberg conjectures’, J. Amer. Math. Soc. 16 (2003), 175–181.Google Scholar
Kowalski, E., Michel, P. and Vanderkam, J., ‘Rankin–Selberg L-functions in the level aspect’, Duke Math. J. 114 (2002), 123–191.10.1215/S0012-7094-02-11416-1Google Scholar
Meurman, T., ‘On the binary additive divisor problem’, in: Turku Sympos. Number Theory, Turku, 1999 (de Gruyter, Berlin, 2001), 223–246.Google Scholar
Munshi, R., ‘Shifted convolution of divisor function $d_{3}$ and Ramanujan $\unicode[STIX]{x1D70F}$-function’, in: The Legacy of Srinivasa Ramanujan, Ramanujan Mathematical Society Lecture Notes Series, 20 (Ramanujan Mathematical Society, Mysore, 2013), 251–260.Google Scholar
Munshi, R., ‘Shifted convolution sums for GL(3) × GL(2)’, Duke Math. J. 162 (2013), 2345–2362.10.1215/00127094-2371416Google Scholar
Pitt, N. J. E., ‘On shifted convolutions of 𝜁3(s) with automorphic L-functions’, Duke Math. J. 77 (1995), 383–406.10.1215/S0012-7094-95-07711-4Google Scholar
Tang, H. C., ‘Shifted convolution sum of d 3 and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc. 92 (2015), 195–204.10.1017/S000497271500043XGoogle Scholar
Topacogullari, B., ‘The shifted convolution sum of divisor functions’, Q. J. Math. 67 (2016), 331–363.10.1093/qmath/haw010Google Scholar
Xi, P., ‘A shifted convolution sum for GL(3) × GL(2)’, Forum Math. 30 (2018), 1013–1027.Google Scholar