Abstract
Let \(\lambda _\phi (n)\) be the \(n\)-th Fourier coefficient of a doubly even and normalized Hecke–Maass cusp form for hyperbolic \(3\)-manifolds. In this paper, we investigate the behavior of summatory functions in the following
-
(i)
the \(j\)-th power sum of \(\lambda _\phi (n)\)
$$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n)^{j}, \end{aligned}$$where \(j\le 8;\)
-
(ii)
the sum of \(\lambda _\phi (n)\) over the sparse sequence \({n^l}\)
$$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l), \end{aligned}$$where \(l\le 4;\)
-
(iii)
the hybrid sum for \(\lambda _\phi (n)\)
$$\begin{aligned} \sum _{N(n)\le x}\lambda _\phi (n^l)^{j}, \end{aligned}$$where \(2\le l\le 4, j=2,\) or \(l=2, j=4.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blomer, V., Brumley, F.: On the Ramanujan conjecture over number fields. Ann. Math. 174, 581–605 (2011)
Chandrasekharan, K., Narasimhan, M.: Functional equations with multiple Gamma factors and the average order of arithmetical functions. Ann. Math. 76, 93–136 (1962)
Iwaniec, H., Kowalski, E.: Analytic number theory. In: American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)
Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations I. Am. J. Math. 103, 499–558 (1981)
Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic forms II. Am. J. Math. 103, 777–815 (1981)
Kaminishi, C.: Estimates of automorphic L-functions in the discriminant-aspect. Proc. Jpn. Acad. Ser. A Math. Sci. 80, 42–46 (2004)
Kim, H.H., Shahidi, F.: Cuspidality of symmetric powers with applications. Duke Math. J. 112, 177–197 (2002)
Kim, H.H., Shahidi, F.: Functorial products for \(GL_2 \times GL_3\) and the symmetric cube for \(GL_2\). Ann. Math. 155, 837–893 (2002)
Kim, H.H.: Functoriality for the exterior square of \(GL_4\) and the symmetric fourth o f \(GL_2\), Appendix 1 by D. Ramakrishnan, Appendix 2 by H. H. Kim and P. Sarnak. J. Am. Math. Soc. 16, 139–183 (2003)
Koyama, S.: \(L^\infty \)-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds. Duke Math. J. 77, 799–817 (1995)
Koyama, S.: Quantum ergodicity of Eisenstein series for arithmetic 3-manifolds. Commun. Math. Phys. 215, 477–486 (2000)
Koyama, S.: The first eigenvalue problem and tensor products of zeta functions. Proc. Jpn. Acad. Ser. A Math. Sci. 80, 35–39 (2004)
Lau, Y.K., Lü, G.S.: Sums of Fourier coefficients of cusp forms. Quart. J. Math. 62, 687–716 (2011)
Lü, G.S.: On averages of Fourier coefficients of Maass cusp forms. Arch. Math. (Basel) 100, 255–265 (2013)
Nakasuji, M.: Generalized Ramanujan conjecture over general imaginary quadratic fields. Forum Math. 24, 85–98 (2012)
Petridis, Y.N., Sarnak, P.: Quantum unique ergodicity for \(SL_2({\cal O}){\backslash } \mathbb{H}_3\) and estimates for L-functions. J. Evol. Equ. 1, 277–290 (2001)
Sarnak, P.: The arithmetic and geometry of some hyperbolic three manifolds. Acta Math. 151, 253–295 (1983)
Shahidi, F.: On certain L-functions. Am. J. Math. 103, 297–355 (1981)
Shahidi, F.: Fourier transforms of intertwining operators and Plancherel measures for \(GL(n)\). Am. J. Math. 106, 67–111 (1984)
Shahidi, F.: Local coefficients as Artin factors for real groups. Duke Math. J. 52, 973–1007 (1985)
Shahidi, F.: A proof of Langlands’ conjecture on Plancherel measures: complementary series for p-adic groups. Ann. Math. 132, 273–330 (1990)
Shimura, G.: On the holomorphy of certain Drichlet series. Proc. Lond. Math. Soc. 31, 79–98 (1975)
Speizer, A.: The theory of binary quadratic forms with coefficients in an arbitrary number field. Dissertation, University of G6ttingen (1909)
Steil, G.: Eigenvalues of the Laplacian for Bianchi groups. In: Emerging Applications of Number Theory, IMA Vol. Math. Appl., vol. 109, pp. 617–641. Springer, New York (1999
Zhao, Y.D.: Certain Dirichlet series attached to automorphic forms over imaginary quadratic fields. Duke Math. J. 72, 695–724 (1993)
Acknowledgments
The authors would like to thank the referee for some helpful comments and suggestions. This work is supported in part by the National Natural Science Foundation of China (11031004, 11171182), IRT1264 and Shandong Province Natural Science Foundation for Distinguished Young Scholars (JQ201102).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by K. Gröchenig.
Rights and permissions
About this article
Cite this article
Jiang, Y., Lü, G. Average behavior of Fourier coefficients of Maass cusp forms for hyperbolic \(3\)-manifolds. Monatsh Math 178, 221–236 (2015). https://doi.org/10.1007/s00605-015-0766-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-015-0766-z