Abstract
Suppose that x is a sufficiently large number and \(j\ge 2\) is any integer. Let \(L(s, \textrm{sym}^j f)\) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL\(_{2}(\mathbb {Z})\). Also, let \(\lambda _{\textrm{sym}^j f}(n)\) be the n-th normalized Dirichlet coefficient of \(L(s, \textrm{sym}^j f)\). In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients \(\lambda _{\textrm{sym}^j f}(n)\) over two sparse sequences of positive integers, which improves previous results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
No data was used for the research described in the article.
References
Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. J. Am. Math. Soc. 30(1), 205–224 (2017)
Deligne, P.: La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. 43(1), 273–307 (1974)
Fomenko, O.M.: Mean value theorems for automorphic \(L\)-functions. St. Petersburg Math. J. 19(5), 853–866 (2008)
Fomenko, O.M.: Identities involving coefficients of automorphic \(L\)-functions. J. Math. Sci. (N.Y.) 133(6), 1749–1755 (2006)
Hua, G.D.: The average behaviour of Hecke eigenvalues over certain sparse sequence of positive integers. Res. Number Theory 8(4), 95, 20 (2022)
Huang, B.R.: Hybrid subconvexity bounds for twisted \(L\)-functions on \(GL(3)\). Sci. China Math. 64(3), 443–478 (2021)
Ivić, A.: Exponent pairs and the zeta function of Riemann. Studia Sci. Math. Hungar. 15, 157–181 (1980)
Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc. Colloq. Publ., vol. 53. Amer. Math. Soc., Providence (2004)
Lao, H.X.: On the fourth moment of coefficients of symmetric square \(L\)-function. Chin. Ann. Math. Ser. B 33(6), 877–888 (2012)
Lau, Y.K., Lü, G.S.: Sums of Fourier coefficients of cusp forms. Q. J. Math. 62(3), 687–716 (2011)
Lin, Y.X., Nunes, R.M., Qi, Z.: Strong subconvexity for self-dual GL(3) \(L\)-functions. Int. Math. Res. Not. IMRN 13, 11453–11470 (2023)
Liu, H.F.: The second moment of the Fourier coefficients of triple product \(L\)-functions. Proc. Indian Acad. Sci. Math. Sci. 133(1), 8–12 (2023)
Liu, H.F.: On the asymptotic distribution of Fourier coefficients of cusp forms. Bull. Braz. Math. Soc. (N.S.) 54(2), 21, 17 (2023)
Luo, S., Lao, H.X., Zou, A.Y.: Asymptotics for the Dirichlet coefficients of symmetric power \(L\)-functions. Acta Arith. 199(3), 253–268 (2021)
Matsumoto, K.: The mean values and the universality of Rankin–Selberg \(L\)-functions. In: Number Theory (Turku, 1999), pp. 201–221. de Gruyter, Berlin (2001)
Perelli, A.: General \(L\)-functions. Ann. Mat. Pura Appl. 130, 287–306 (1982)
Ramachandra, K., Sankaranarayanan, A.: Notes on the Riemann zeta-function. J. Indian Math. Soc. 57(1–4), 67–77 (1991)
Sharma, A., Sankaranarayanan, A.: On the average behavior of the Fourier coefficients of \(j\)th symmetric power \(L\)-function over certain sequences of positive integers. Czech. Math. J. 73(3), 885–901 (2023)
Sharma, A., Sankaranarayanan, A.: Higher moments of the Fourier coefficients of symmetric square \(L\)-functions on certain sequence. Rend. Circ. Mat. Palermo (2) 72(2), 1399–1416 (2023)
Sharma, A., Sankaranarayanan, A.: Average behavior of the Fourier coefficients of the symmetric square \(L\)-function over some sequence of integer. Integers 22, A74, 17 (2022)
Sharma, A., Sankaranarayanan, A.: Discrete mean square of the coefficients of symmetric square \(L\)-functions on certain sequence of positive numbers. Res. Number Theory 8(1), 19, 13 (2022)
Tang, H.C.: Estimates for the Fourier coefficients of symmetric square \(L\)-functions. Arch. Math. (Basel) 100(2), 123–130 (2013)
Xu, C.R.: General asymptotic formula of Fourier coefficients of cusp forms over sum of two squares. J. Number Theory 236, 214–229 (2022)
Zhai, S.: Average behavior of Fourier coefficients of cusp forms over sum of two squares. J. Number Theory 133(11), 3862–3876 (2013)
Acknowledgements
The authors would like to thank the referees for many useful comments on the manuscript. This work is supported by the National Natural Science Foundation of China (Grant Nos. 12171286 and 11801328).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this article.
Additional information
Communicated by Mohammad Reza Koushesh.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, H., Yang, X. The Average Behaviors of the Fourier Coefficients of j-th Symmetric Power L-Function over Two Sparse Sequences of Positive Integers. Bull. Iran. Math. Soc. 50, 14 (2024). https://doi.org/10.1007/s41980-023-00850-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s41980-023-00850-z