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

DOI:
https://doi.org/10.1090/S0002-9939-09-09855-4

Published electronically:
March 12, 2009

MathSciNet review:
2497466

Full-text PDF Free Access

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 $\varepsilon >0$, \[ \sum _{n \leq x}\lambda ^2(n^j)=c_{j-1} x+O\left (x^{1-\frac {2}{(j+1)^2+2}+\varepsilon }\right ),\] where $j=2,3,4.$

- 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**, DOI https://doi.org/10.1155/S1073792804132455 - Pierre Deligne,
*La conjecture de Weil. I*, Inst. Hautes Études Sci. Publ. Math.**43**(1974), 273–307 (French). MR**340258** - 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. (POMI)**276**(2001), no. Anal. Teor. Chisel i Teor. Funkts. 17, 300–311, 354 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.)**118**(2003), no. 1, 4910–4917. MR**1850374**, DOI https://doi.org/10.1023/A%3A1025537019956 - Stephen Gelbart and Hervé Jacquet,
*A relation between automorphic representations of ${\rm GL}(2)$ and ${\rm GL}(3)$*, Ann. Sci. École Norm. Sup. (4)**11**(1978), no. 4, 471–542. MR**533066** - Aleksandar 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) Mosk. Gos. Univ. im. Lomonosova, Mekh.-Mat. Fak., Moscow, 2002, pp. 92–97. MR**1985942** - Henryk Iwaniec,
*Topics in classical automorphic forms*, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR**1474964** - Henryk Iwaniec and Emmanuel Kowalski,
*Analytic number theory*, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR**2061214** - Henry H. Kim,
*Functoriality for the exterior square of ${\rm GL}_4$ and the symmetric fourth of ${\rm GL}_2$*, J. Amer. Math. Soc.**16**(2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR**1937203**, DOI https://doi.org/10.1090/S0894-0347-02-00410-1 - Henry H. Kim and Freydoon Shahidi,
*Functorial products for ${\rm GL}_2\times {\rm GL}_3$ and the symmetric cube for ${\rm GL}_2$*, Ann. of Math. (2)**155**(2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart. MR**1923967**, DOI https://doi.org/10.2307/3062134 - Henry H. Kim and Freydoon Shahidi,
*Cuspidality of symmetric powers with applications*, Duke Math. J.**112**(2002), no. 1, 177–197. MR**1890650**, DOI https://doi.org/10.1215/S0012-9074-02-11215-0 - Yuk-Kam Lau and Jie Wu,
*A density theorem on automorphic $L$-functions and some applications*, Trans. Amer. Math. Soc.**358**(2006), no. 1, 441–472. MR**2171241**, DOI https://doi.org/10.1090/S0002-9947-05-03774-8 - G. S. Lü, On sums of Fourier coefficients of cusp forms over sparse sequences, to appear in Science in China, Ser. A.
- 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.
- R. A. Rankin,
*Sums of cusp form coefficients*, Automorphic forms and analytic number theory (Montreal, PQ, 1989) Univ. Montréal, Montreal, QC, 1990, pp. 115–121. MR**1111014** - A. Sankaranarayanan,
*Fundamental properties of symmetric square $L$-functions. I*, Illinois J. Math.**46**(2002), no. 1, 23–43. MR**1936073** - A. Sankaranarayanan,
*On a sum involving Fourier coefficients of cusp forms*, Liet. Mat. Rink.**46**(2006), no. 4, 565–583 (English, with English and Lithuanian summaries); English transl., Lithuanian Math. J.**46**(2006), no. 4, 459–474. MR**2320364**, DOI https://doi.org/10.1007/s10986-006-0042-y - Atle Selberg,
*Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist*, Arch. Math. Naturvid.**43**(1940), 47–50 (German). MR**2626** - Freydoon Shahidi,
*Third symmetric power $L$-functions for ${\rm GL}(2)$*, Compositio Math.**70**(1989), no. 3, 245–273. MR**1002045**

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

Email:
laohuixue@sina.com

**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

Email:
sank@math.tifr.res.in, ayyadurai.sankaranarayanan@uni-ulm.de

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.