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Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m, ℤ)

Published online by Cambridge University Press:  25 April 2016

YUJIAO JIANG ,
GUANGSHI LÜ  and
XIAOFEI YAN
YUJIAO JIANG
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China. e-mails: [email protected]
GUANGSHI LÜ
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China. e-mails: [email protected]
XIAOFEI YAN
Affiliation:
School of Mathematics, Shandong Normal University, Jinan, Shandong 250014, China. e-mails: [email protected]
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Abstract

Let F(z) be a Hecke–Maass form for SL(m, ℤ) with m ⩽ 3, or be the symmetric power lift of a Hecke–Maass form for SL(2, ℤ) if m = 4, 5 and let AF (n, 1, . . ., 1) be the coefficients of L-function attached to F. We establish

$$\sum_{q\leq Q}\max_{(a,q)=1}\max_{y\leq x}\left|\sum_{n\leq y \atop n\equiv a\bmod q}A_F(n,1, \dots, 1)\Lambda(n)\right| \ll x\log^{-A}x,$$
where Q = x ϑ−ϵ with some ϑ > 0, the implied constant depends on F, A, ϵ.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 2 , September 2016 , pp. 339 - 356
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Banks, W. D. Twisted symmetric-square L-functions and the nonexistence of siegel zeros on gl (3). Duke Math. J. 87 (2) (1997), 343–354.CrossRefGoogle Scholar
[2] Bump, D. Automorphic Forms and Representations, volume 55 (Cambridge University Press, 1998).Google Scholar
[3] Chandrasekharan, K. and Narasimhan, R. Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. of Math. (2), 76 (1962), 93–136.Google Scholar
[4] Friedlander, J. B. and Iwaniec, H. Summation formulae for coefficients of L-functions. Canad. J. Math. 57 (3) (2005), 494–505.CrossRefGoogle Scholar
[5] Goldfeld, D. Automorphic Forms and L-Functions for the Group GL(n, R) Cambridge Studies in Advanced Math. vol 99 (Cambridge University Press, Cambridge, 2006). With an appendix by Kevin A. Broughan.Google Scholar
[6] Hoffstein, J. and Ramakrishnan, D. Siegel zeros and cusp forms. Internat. Math. Res. Notices. (6) (1995), 279–308.CrossRefGoogle Scholar
[7] Iwaniec, H. and Kowalski, E. Analytic Number Theory. Amer. Math. Soc. Colloq. Publ. vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
[8] Kim, H. H. Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 . J. Amer. Math. Soc. 16 (1) (2003), 139–183. With appendix 1 by D. Ramakrishnan and appendix 2 by K. and P. Sarnak.CrossRefGoogle Scholar
[9] Kim, H. H. and Shahidi, F. Functorial products for gl2×gl3 and the symmetric cube for gl2. Annals of math. 155 (3) (2002), 837–893.Google Scholar
[10] Kim, H. H., Shahidi, F. Cuspidality of symmetric powers with applications. Duke Math. J. 112 (1) (2002), 177–197.Google Scholar
[11] Lü, G. On sums involving coefficients of automorphic L-functions. Proc. Amer. Math. Soc. 137 (9) (2009), 2879–2887.Google Scholar
[12] Lü, G. On averages of Fourier coefficients of Maass cusp forms. Arch. Math. (Basel) 100 (3) (2013), 255–265.Google Scholar
[13] Molteni, G. L-functions: Siegel-type theorems and structure theorems. PhD thesis. University of Milan (1999).Google Scholar
[14] Perelli, A. Exponential sums and mean-value theorems connected with Ramanujan's τ-function. In Seminar on Number Theory (Talence, 1983/1984), pages Exp. No. 25, 9. (Univ. Bordeaux I, Talence, 1984).Google Scholar
[15] Smith, R. A. The average order of a class of arithmetic functions over arithmetic progressions with applications to quadratic forms. J. Reine Angew. Math. 317 (1980), 74–87.Google Scholar
[16] Vaughan, R. An elementary method in prime number theory. Acta Arithmetica 37 (1) (1980), 111–115.CrossRefGoogle Scholar