Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

Ω-RESULT ON COEFFICIENTS OF AUTOMORPHIC L-FUNCTIONS OVER SPARSE SEQUENCES

  • LAO, HUIXUE (Department of Mathematics Shandong Normal University) ;
  • WEI, HONGBIN (Department of Mathematics Shandong Normal University)
  • Received : 2014.10.26
  • Published : 2015.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let ${\lambda}_f(n)$ denote the n-th normalized Fourier coefficient of a primitive holomorphic form f for the full modular group ${\Gamma}=SL_2({\mathbb{Z}})$. In this paper, we are concerned with ${\Omega}$-result on the summatory function ${\sum}_{n{\leqslant}x}{\lambda}^2_f(n^2)$, and establish the following result ${\sum}_{\leqslant}{\lambda}^2_f(n^2)=c_1x+{\Omega}(x^{\frac{4}{9}})$, where $c_1$ is a suitable constant.

Keywords

References

  1. P. Deligne, La Conjecture de Weil, Inst. Hautes Etudes Sci. Pul. Math. 43 (1974), 29-39.
  2. O. M. Fomenko, Identities involving coefficients of automorphic L-functions, J. Math. Sci. 133 (2006), 1749-1755. https://doi.org/10.1007/s10958-006-0086-x
  3. S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. 11 (1978), no. 4, 471-552. https://doi.org/10.24033/asens.1355
  4. J. L. Hafner and A. Ivic, On sums of Fourier coefficients of cusp forms, Enseign. Math. 35 (1989), no. 3-4, 375-382.
  5. A. Ivic, On sums of Fourier coefficients of cusp form, IV International Conference "Modern Problems of Number Theory and its Applications": current problems, part II(Russia) (Tula, 2001), 92-97, Mosk. Gos. Univ. im. Lomonosoua, Mekh. Mat. Fak., Moscow, 2002.
  6. H. Iwaniec, Topics in Classical Automorphic Forms, American Mathematical Society, Providence, RI, 1997.
  7. H. Kim, Functoriality for the exterior square of $GL_4$ and symmetric fourth of $GL_2$, Appendix 1 by D. Ramakrishnan, Appendix 2 by H. Kim and P. Sarnak, J. Amer. Math. Soc. 16 (2003), no. 1, 139-183. https://doi.org/10.1090/S0894-0347-02-00410-1
  8. H. Kim and F. Shahidi, Functorial products for $GL_2{\time}GL_3$ and functorial symmetric cube for $GL_2$, With an appendix by C. J. Bushnell and G. Henniart, Ann. of Math. 155 (2002), no. 3, 837-893. https://doi.org/10.2307/3062134
  9. M. Kuhleitner and W. G. Nowak, An omega theorem for a class of arithmetic functions, Math. Nachr. 165 (1994), 79-98. https://doi.org/10.1002/mana.19941650107
  10. H. X. Lao and A. Sankaranarayanan, The average behavior of Fourier coefficients of cusp forms over sparse sequences, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2557-2565. https://doi.org/10.1090/S0002-9939-09-09855-4
  11. Y.-K. Lau and G. S. Lu, Sums of Fourier coefficients of cusp forms, Quart. J. Math. Oxford Ser. 62 (2011), no. 3, 687-716. https://doi.org/10.1093/qmath/haq012
  12. Y.-K. Lau, G. S. Lu, and J. Wu, Integral power sums of Hecke eigenvalues, Acta Arith. 150 (2011), no. 2, 193-207. https://doi.org/10.4064/aa150-2-7
  13. Y.-K. Lau and J. Wu, A density theorem on automorphic L-functions and some applications, Trans. Amer. Math. Soc. 359 (2006), no. 1, 441-472.
  14. G. S. Lu, On an open problem of Sankaranarayanan, Sci. China Math. 53 (2010), no. 5, 1319-1324. https://doi.org/10.1007/s11425-009-0183-7
  15. A. Mukhopadhyay and K. Srinivas, A zero density estimate for the Selberg class, Int. J. Number Theory 3 (2007), no. 2, 263-273. https://doi.org/10.1142/S1793042107000894
  16. R. A. Rankin, Contributions to the theory of Ramanujan's function ${\tau}$(n) and similar arithemtical functions I. The zeros of the function ${\Sigma}_{n=1}^{\infty}{\tau}(n)/n^s$ on the line Rs = 13/2. II. The order of the Fourier coefficients of the integral modular forms, Proc. Cambridge Phil. Soc. 35 (1939), 351-372. https://doi.org/10.1017/S0305004100021095
  17. R. A. Rankin, Sums of powers of cusp form coefficients, Math. Ann. 63 (1983), no. 2, 227-236.
  18. R. A. Rankin, Sums of powers of cusp form coefficients II, Math. Ann. 272 (1985), no. 4, 593-600. https://doi.org/10.1007/BF01455869
  19. R. A. Rankin, Sums of cusp form coefficients, Automorphic forms and analytic number theory (Montreal, PQ, 1989), 115-121, Univ. Montreal, Montreal, QC, 1990.
  20. A. Sankaranarayanan, On a sum involving Fourier coefficients of cusp forms, Lithuanian Math. J. 46 (2006), no. 4, 459-474. https://doi.org/10.1007/s10986-006-0042-y
  21. A. Selberg, Bemerkungen uber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid 43 (1940), 47-50.
  22. F. Shahidi, Third symmetric power L-functions for GL(2), Compos. Math. 70 (1989), no. 3, 245-273.