The general two-dimensional divisor problems involving Hecke eigenvalues

Jing Huang; Taiyu Li; Huafeng Liu; Fuxia Xu; Jing Huang; Taiyu Li; Huafeng Liu; Fuxia Xu

doi:10.3934/math.2022356

AIMS Mathematics

2022, Volume 7, Issue 4: 6396-6403. doi: 10.3934/math.2022356

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The general two-dimensional divisor problems involving Hecke eigenvalues

1.
School of Mathematics and Statistics, Shandong Normal University, 88 Wenhua Donglu, Jinan, Shandong 250014, China
2.
School of Mathematics and Statistics, Shandong University, 180 Wenhua Xilu, Weihai, Shandong 264209, China

Received: 17 November 2021 Revised: 31 December 2021 Accepted: 14 January 2022 Published: 19 January 2022
MSC : 11N37, 11A25

We consider the general two-dimensional divisor problems involving Hecke eigenvalues, and are able to improve the previous results in this direction.
- Hecke eigenvalue,
- divisor problem,
- Fourier coefficients
Citation: Jing Huang, Taiyu Li, Huafeng Liu, Fuxia Xu. The general two-dimensional divisor problems involving Hecke eigenvalues[J]. AIMS Mathematics, 2022, 7(4): 6396-6403. doi: 10.3934/math.2022356

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Abstract

We consider the general two-dimensional divisor problems involving Hecke eigenvalues, and are able to improve the previous results in this direction.

References

[1]	J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc., 30 (2017), 205–224. https://doi.org/10.1090/jams/860 doi: 10.1090/jams/860
[2]	J. Bourgain, N. Watt, Mean square of zeta function, circle problem and divisor problem revisited, arXiv Preprint, 2017. Available from: https://arXiv.org/abs/1709.04340.
[3]	P. Deligne, La conjecture de Weil. I, Publ. Math. Inst. Hautes Études Sci., 43 (1974), 273–307. https://doi.org/10.1007/BF02684373
[4]	P. Deligne, La conjecture de Weil. II, Publ. Math. Inst. Hautes Études Sci., 52 (1980), 137–252. https://doi.org/10.1007/BF02684780
[5]	S. Gelbart, H. Jacquet, A relation between automorphic representations of $GL(2)$ and $GL(3)$, Ann. Sci. École Norm. Sup., 11 (1978), 471–542. https://doi.org/10.24033/asens.1355 doi: 10.24033/asens.1355
[6]	E. Hecke, Theorie der Eisensteinsche Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Sem. Hamburg, 5 (1927), 199–224. https://doi.org/10.1007/BF02952521 doi: 10.1007/BF02952521
[7]	J. Huang, H. F. Liu, F. X. Xu, Two-dimension divisor problems related to symmetric $L$-function, Symmetry, 13 (2021), 1–12. https://doi.org/10.3390/sym13020359 doi: 10.3390/sym13020359
[8]	A. Ivić, The Riemann zeta-function: The theory of the Riemann zeta-function with applications, New York: John Wiley and Sons, 1985.
[9]	H. Iwaniec, Topics in classical automorphic forms, Providence, Rhode lsland: American Mathematical Society, 1997. https://doi.org/10.1090/gsm/017
[10]	H. Iwaniec, E. Kowalski, Analytic number theory, Providence, Rhode lsland: American Mathematical Society, 2004. https://doi.org/10.1090/coll/053
[11]	Y. J. Jiang, G. S. Lü, X. F. Yan, Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for $SL(m, Z)$, Math. Proc. Cambridge Philos. Soc., 161 (2016), 339–356. https://doi.org/10.1017/S030500411600027X doi: 10.1017/S030500411600027X
[12]	S. Kanemitsu, A. Sankaranarayanan, Y. Tanigawa, A mean value theorem for Dirichlet series and a general divisor problem, Monatsh. Math., 136 (2002), 17–34. https://doi.org/10.1007/s006050200031 doi: 10.1007/s006050200031
[13]	E. Krätzel, Teilerprobleme in drei Dimensionen, Math. Nachr., 42 (1969), 275–288. https://doi.org/10.1002/mana.19690420408 doi: 10.1002/mana.19690420408
[14]	H. X. Lao, The cancellation of Fourier coefficients of cusp forms over different sparse sequences, Acta Math. Sin. Engl. Ser., 29 (2013), 1963–1972. https://doi.org/10.1007/s10114-013-2706-y doi: 10.1007/s10114-013-2706-y
[15]	H. X. Lao, Mean value of Dirichlet series coefficients of Rankin-Selberg $L$-functions, Lith. Math. J., 57 (2017), 351–358. https://doi.org/10.1007/s10986-017-9365-0 doi: 10.1007/s10986-017-9365-0
[16]	H. F. Liu, R. Zhang, Some problems involving Hecke eigenvalues, Acta. Math. Hungar., 159 (2019), 287–298. https://doi.org/10.1007/s10474-019-00913-w doi: 10.1007/s10474-019-00913-w
[17]	G. S. Lü, On general divisor problems involving Hecke eigenvalues, Acta. Math. Hungar., 135 (2012), 148–159. https://doi.org/10.1007/s10474-011-0150-y doi: 10.1007/s10474-011-0150-y
[18]	R. M. Nunes, On the subconvexity estimate for self-dual $GL(3)$ $L$-functions in the $t$-aspect, arXiv Preprint, 2017. Available from: https://arXiv.org/abs/1703.04424.
[19]	A. Perelli, General $L$-functions, Ann. Mat. Pura Appl., 130 (1982), 287–306. https://doi.org/10.1007/BF01761499
[20]	R. A. Rankin, Contributions to the theory of Ramanujan's function $\tau(n)$ and similar arithemtical functions, Math. Proc. Cambridge Philos. Soc., 35 (1939), 357–372. https://doi.org/10.1017/S0305004100021101 doi: 10.1017/S0305004100021101
[21]	R. A. Rankin, Sums of cusp form coefficients, In: Automorphic forms and analytic number theory, Université de Montréal, 1990,115–121.
[22]	A. Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid., 43 (1940), 47–50.
[23]	J. Wu, Power sums of Hecke eigenvalues and application, Acta Arith., 137 (2009), 333–344. https://doi.org/10.4064/aa137-4-3 doi: 10.4064/aa137-4-3
[24]	D. Y. Zhang, Y. K. Lau, Y. N. Wang, Remark on the paper "On products of Fourier coefficients of cusp forms", Arch. Math., 108 (2017), 263–269. https://doi.org/10.1007/s00013-016-0996-x doi: 10.1007/s00013-016-0996-x
[25]	D. Y. Zhang, Y. N. Wang, Higher-power moments of Fourier coefficients of holomorphic cusp forms for the congruence subgroup $\Gamma_0(N)$, Ramanujan J., 47 (2018), 685–700. https://doi.org/10.1007/s11139-018-0051-6 doi: 10.1007/s11139-018-0051-6

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