Mean-square estimate of automorphic L-functions

Weili YAO

doi:10.1007/s11464-020-0817-1

Front. Math. China ›› 2020, Vol. 15 ›› Issue (1) : 205 -213. DOI: 10.1007/s11464-020-0817-1

RESEARCH ARTICLE

Mean-square estimate of automorphic L-functions

Weili YAO ^†

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Abstract

Let f be a holomorphic Hecke cusp form with even integral weight $k ≥ 2$ for the full modular group, and let $χ$ be a primitive Dirichlet character modulo q. Let $L f (s, χ)$ be the automorphic L-function attached to f and $χ$ . We study the mean-square estimate of $L f (s, χ)$ and establish an asymptotic formula.

Keywords

Automorphic L-function / cusp form / Fourier coe_cient

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Weili YAO. Mean-square estimate of automorphic L-functions. Front. Math. China, 2020, 15(1): 205-213 DOI:10.1007/s11464-020-0817-1

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References

Publishing order | Descend order by publishing year | Descend order by cited within

[1]	Blomer V. Shifted convolution sums and subconvexity bounds for automorphic L-functions. Int Math Res Not IMRN, 2004, 73: 3905–3926

[2]	Blomer V, Harcos G. Hybrid bounds for twisted L-functions. J Reine Angew Math, 2008, 621: 53–79

[3]	Deligne P. Formes modulaires et représentations l-adiques. In: Séminaire Bourbaki, Vol. 1968/69, Exposés 347–363. Lecture Notes in Math, Vol 179. Berlin: Springer, 1971, 139–172

[4]	Deligne P. La conjecture de Weil I. Publ Math Inst Hautes Études Sci, 1974, 43: 273–307

[5]	Duke W, Friedlander J B, Iwaniec H. Bounds for automorphic L-functions. Invent Math, 1993, 112: 1–8

[6]	Hafner J L, Ivić A. On sums of Fourier coefficients of cusp forms. Enseign Math, 1989, 35: 375–382

[7]	Harcos G. An additive problem in the Fourier coefficients of cusp forms. Math Ann, 2003, 326: 347–365

[8]	Iwaniec H. Topics in Classical Automorphic Forms. Grad Stud Math, Vol 17. Providence: Amer Math Soc, 1997

[9]	Lau Y-K, Lü G S. Sums of Fourier coefficients of cusp forms. Quart J Math, 2011, 62: 687–716

[10]	Rankin R A. Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions II. The order of the Fourier coefficients of the integral modular forms. Proc Cambridge Philos Soc, 1939, 35: 351–372

[11]	Sarnak P. Estimates for Rankin-Selberg L-functions and quantum unique ergodicity. J Funct Anal, 2001, 184: 419–453

[12]	Selberg A. Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist. Arch Math Naturvid, 1940, 43: 47–50

[13]	Shimura G. Introduction to the Arithmetic Theory of Automorphic Functions. Princeton: Princeton Univ Press, 1971

[14]	Yi Y, Zhang W P. On the 2k-th power mean of inversion of L-functions with the weight of the Gauss sum. Acta Math Sin (Engl Ser), 2004, 20: 175–180