Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The average behavior of Fourier coefficients of cusp forms over sparse sequences

Authors: Huixue Lao and Ayyadurai Sankaranarayanan
Journal: Proc. Amer. Math. Soc. 137 (2009), 2557-2565
MSC (2000): Primary 11F30, 11F11, 11F66
Published electronically: March 12, 2009
MathSciNet review: 2497466
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \lambda(n)$ be the $ n$th normalized Fourier coefficient of a holomorphic Hecke eigenform $ f(z)\in S_{k}(\Gamma)$. In this paper we are interested in the average behavior of $ \lambda^2(n)$ over sparse sequences. By using the properties of symmetric power $ L$-functions and their Rankin-Selberg $ L$-functions, we are able to establish that for any $ \varepsilon>0$,

$\displaystyle \sum_{n \leq x}\lambda^2(n^j)=c_{j-1} x+O\left(x^{1-\frac{2}{(j+1)^2+2}+\varepsilon}\right),$

where $ j=2,3,4.$

References [Enhancements On Off] (What's this?)

  • 1. J. Cogdell and P. Michel, On the complex moments of symmetric power $ L$-functions at $ s=1$, Int. Math. Res. Not., 31(2004), 1561-1617. MR 2035301 (2005f:11094)
  • 2. P. Deligne, La Conjecture de Weil, I, Inst. Hautes Études Sci. Publ. Math., 43(1974), 273-307.MR 0340258 (49:5013)
  • 3. O. M. Fomenko, On the behavior of automorphic $ L$-functions at the center of the critical strip, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMl), 276(2001), Anal. Teor. Chisel i Teor. Funkts., 17, 300-311, 354; translation in J. Math. Sci. (N. Y.), 118(2003), 4910-4917. MR 1850374 (2002g:11060)
  • 4. S. Gelbart and H. Jacquet, A relation between automorphic representations of GL$ (2)$ and GL$ (3)$, Ann. Sci. École Norm. Sup., 11(1978), 471-542. MR 533066 (81e:10025)
  • 5. A. Ivić, On sums of Fourier coefficients of cusp forms, IV International Conference ``Modern Problems of Number Theory and Its Applications'': Current Problems, Part II (Russian) (Tula, 2001), 92-97, Mosk. Gos. Univ. im. Lomonosova, Mekh.-Mat. Fak., Moscow, 2002. MR 1985942 (2004d:11030)
  • 6. H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Math., vol. 17, Amer. Math. Soc., Providence, RI, 1997. MR 1474964 (98e:11051)
  • 7. H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloquium Publ. 53, Amer. Math. Soc., Providence, RI, 2004. MR 2061214 (2005h:11005)
  • 8. H. Kim, Functoriality for the exterior square of $ {GL}_4$ and the symmetric fourth of $ {GL}_2$, Appendix $ 1$ by D. Ramakrishnan, Appendix $ 2$ by H. Kim and P. Sarnak, J. Amer. Math. Soc., 16(2003), 139-183. MR 1937203 (2003k:11083)
  • 9. H. Kim and F. Shahidi, Functorial products for $ {GL}_2 \times {GL}_3$ and the symmetric cube for $ {GL}_2$ (with an appendix by C. J. Bushnell and G. Henniart), Ann. of Math. (2), 155(2002), 837-893. MR 1923967 (2003m:11075)
  • 10. H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J., 112(2002), 177-197. MR 1890650 (2003a:11057)
  • 11. Y.-K. Lau and J. Wu, A density theorem on automorphic $ L$-functions and some applications, Trans. Amer. Math. Soc., 358(2006), 441-472. MR 2171241 (2006g:11097)
  • 12. G. S. Lü, On sums of Fourier coefficients of cusp forms over sparse sequences, to appear in Science in China, Ser. A.
  • 13. R. A. Rankin, Contributions to the theory of Ramanujan's function $ \tau(n)$ and similar arithmetical functions, II. The order of the Fourier coefficients of the integral modular forms, Proc. Cambridge Phil. Soc., 35(1939), 357-372.
  • 14. R. A. Rankin, Sums of cusp form coefficients. Automorphic forms and analytic number theory (Montreal, PQ, 1989), 115-121, Univ. Montreal, Montreal, QC, 1990. MR 1111014 (92g:11042)
  • 15. A. Sankaranarayanan, Fundamental properties of symmetric square $ L$-functions, I, Illinois J. Math., 46(2002), 23-43. MR 1936073 (2004a:11038)
  • 16. A. Sankaranarayanan, On a sum involving Fourier coefficients of cusp forms, Lithuanian Mathematical Journal, 46(2006), 459-474. MR 2320364 (2008c:11068)
  • 17. A. Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid., 43(1940), 47-50. MR 0002626 (2:88a)
  • 18. F. Shahidi, Third symmetric power $ L$-functions for GL$ (2)$, Compos. Math., 70(1989), 245-273. MR 1002045 (90m:11081)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F30, 11F11, 11F66

Retrieve articles in all journals with MSC (2000): 11F30, 11F11, 11F66

Additional Information

Huixue Lao
Affiliation: Department of Mathematics, Shandong Normal University, Jinan Shandong, 250014, People’s Republic of China

Ayyadurai Sankaranarayanan
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400 005, India – and – Institute for Number Theory and Probability Theory, University of Ulm, D-89069, Ulm, Germany

Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function, Rankin-Selberg $L$-function
Received by editor(s): October 17, 2008
Published electronically: March 12, 2009
Additional Notes: This work is supported by the National Natural Science Foundation of China (Grant No. 10701048)
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society