Average behavior of Fourier coefficients of cusp forms
HTML articles powered by AMS MathViewer
- by Guangshi Lü PDF
- Proc. Amer. Math. Soc. 137 (2009), 1961-1969 Request permission
Abstract:
Let $a_0(n)$ and $b_0(n)$ be the normalized Fourier coefficients of the two holomorphic Hecke eigenforms $f(z)\in S_{2k}(\Gamma )$ and $\varphi (z)\in S_{2l}(\Gamma )$ respectively. In 1999, Fomenko studied the following average sums of $a_0(n)$ and $b_0(n)$: \[ \sum _{n \leq x}a_0(n)^3, \quad \sum _{n \leq x}a_0(n)^2b_0(n), \quad \sum _{n \leq x}a_0(n)^2b_0(n)^2, \quad \sum _{n \leq x}a_0(n)^4. \] In this paper, we are able to improve on Fomenko’s results.References
- Daniel Bump and David Ginzburg, Symmetric square $L$-functions on $\textrm {GL}(r)$, Ann. of Math. (2) 136 (1992), no. 1, 137–205. MR 1173928, DOI 10.2307/2946548
- K. Chandrasekharan and Raghavan Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93–136. MR 140491, DOI 10.2307/1970267
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
- O. M. Fomenko, Fourier coefficients of parabolic forms, and automorphic $L$-functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 237 (1997), no. Anal. Teor. Chisel i Teor. Funkts. 14, 194–226, 231 (Russian, with Russian summary); English transl., J. Math. Sci. (New York) 95 (1999), no. 3, 2295–2316. MR 1691291, DOI 10.1007/BF02172473
- Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066, DOI 10.24033/asens.1355
- Dorian Goldfeld, Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662, DOI 10.1017/CBO9780511542923
- H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
- Hervé Jacquet and Joseph Shalika, Rankin-Selberg convolutions: Archimedean theory, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 125–207. MR 1159102
- Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017
- Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
- Carlos J. Moreno and Freydoon Shahidi, The fourth moment of Ramanujan $\tau$-function, Math. Ann. 266 (1983), no. 2, 233–239. MR 724740, DOI 10.1007/BF01458445
- 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.
- Freydoon Shahidi, Third symmetric power $L$-functions for $\textrm {GL}(2)$, Compositio Math. 70 (1989), no. 3, 245–273. MR 1002045
- Goro Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31 (1975), no. 1, 79–98. MR 382176, DOI 10.1112/plms/s3-31.1.79
Additional Information
- Guangshi Lü
- Affiliation: Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China
- Email: gslv@sdu.edu.cn
- Received by editor(s): May 30, 2008
- Received by editor(s) in revised form: August 28, 2008
- Published electronically: December 30, 2008
- Additional Notes: This work was supported by the National Natural Science Foundation of China (Grant No. 10701048).
- Communicated by: Ken Ono
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1961-1969
- MSC (2000): Primary 11F30, 11F11, 11F66
- DOI: https://doi.org/10.1090/S0002-9939-08-09741-4
- MathSciNet review: 2480277