Abstract
Let λf(n), σ(n) and ϕ(n) denote nth Fourier coefficient of the primitive holomorphic cusp form f, the sum-of-divisors function, and the Euler totient function, respectively. In this paper, we first refine related previous results on the power sum
Funding statement: This work is supported by the Excellent Youth Innovation Team Plan of Shandong Higher Education Institutions (Grant No. 2023KJ197), the National Natural Science Foundation of China (Grant Nos. 12171286 and 11801328) and the Natural Science Foundation of Shandong Province (Grant No. ZR2024MA053).
Communicated by István Gaál
References
[1] Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), 205–224.10.1090/jams/860Search in Google Scholar
[2] Cui, S.: The Average Estimates for Fourier Coefficients of Holomorphic Cusp Forms, Master Thesis, Shandong Normal University, 2018.Search in Google Scholar
[3] Deligne, P.: La Conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. 43 (1973), 273–307.10.1007/BF02684373Search in Google Scholar
[4] Fomenko, O. M.: Fourier coefficients of cusp forms and automorphic L-functions, J. Math. Sci. 95 (1997), 1072–3374.10.1007/BF02172473Search in Google Scholar
[5] Good, A.: The square mean of Dirichlet series associated with cusp forms, Mathematika 29 (1982), 278–295.10.1112/S0025579300012377Search in Google Scholar
[6] Gelbart, S.—Jacquet, H.: A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Ecole Norm. Sup. 11 (1978), 471–542.10.24033/asens.1355Search in Google Scholar
[7] Hardy, G. H.—Wright, E. M.: An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 1979.Search in Google Scholar
[8] Heath-Brown, D. R.: The growth rate of the Dedekind zeta-function on the critical line, Acta Arith. 49 (1988), 323–339.10.4064/aa-49-4-323-339Search in Google Scholar
[9] Ivić, A.: Exponent pairs and the zeta function of Riemann, Studia Sci. Math. Hung. 15 (1980), 157–181.Search in Google Scholar
[10] Ivić, A.: On zeta-functions associated with Fourier coefficients of cusp forms. In: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), 1992, 231–246.Search in Google Scholar
[11] Jiang, Y. J.—Lü, G. S.: On the higher mean over arithmetic progressions of Fourier coefficients of cusp form, Acta Arith. 166 (2014), 231–252.10.4064/aa166-3-2Search in Google Scholar
[12] Kuan, C. I.: Hybrid bounds on twisted L-functions associated with modular forms, J. Number Theory 189 (2018), 380–416.10.1016/j.jnt.2018.02.002Search in Google Scholar
[13] Kim, H. H.: Functoriality for the exterior square of GL4 and symmetric fourth of GL2 (with appendix 1 by D. Ramakrishnan and appendix 2 by H. H. Kim and P. Sarnak), J. Amer. Math. Soc. 16 (2003), 139–183.10.1090/S0894-0347-02-00410-1Search in Google Scholar
[14] Kim, H. H.—Shahidi, F.: Functorial products for GL2 × GL3 and the symmetric cube for GL2 (with an appendix by C. J. Bushnell and G. Henniart), Ann. Math. 155 (2002), 837–893.10.2307/3062134Search in Google Scholar
[15] Lao, H. X.—Luo, S.: Sign changes and non-vanishing of Fourier coefficients of holomorphic cusp forms, Rocky Mountain J. Math. 51 (2021), 1701–1714.10.1216/rmj.2021.51.1701Search in Google Scholar
[16] Lau, Y. K.—Lü, G. S.—Wu, J.: Integral power sums of Hecke eigenvalues, Acta Arith. 150 (2011), 193–207.10.4064/aa150-2-7Search in Google Scholar
[17] Liu, H. F.: On the asymptotic distribution of Fourier coefficients of cusp forms, Bull. Braz. Math. Soc. (N. S.) 54 (2023), Art. No. 21.10.1007/s00574-023-00335-xSearch in Google Scholar
[18] Liu, H. F.: Notes on general divisor problems related to Maass cusp forms, Period. Math. Hungar. 86 (2023), 552–563.10.1007/s10998-022-00490-6Search in Google Scholar
[19] Lin, Y. X.—Nunes, R.—Qi, Z.: Strong subconvexity for self-dual GL3 L-functions, Int. Math. Res. Notices. 2023 (2023), 11453–11470.10.1093/imrn/rnac153Search in Google Scholar
[20] Lü, G. S.: The sixth and eighth moments of Fourier coefficients of cusp forms, J. Number Theory 129 (2009), 2790–2800.10.1016/j.jnt.2009.01.019Search in Google Scholar
[21] Lü, G. S.: Uniform estimates for sums of Fourier coefficients of cusp forms, Acta Math. Hungar. 124 (2009), 83–97.10.1007/s10474-009-8153-7Search in Google Scholar
[22] Manski, S.—Mayle, J.—Zbacnik, N.: The asymptotic distribution of a hybrid arithmetic function, Integers 15 (2015), Art. No. A28.Search in Google Scholar
[23] Matsumoto, K.: The mean values and the universality of Rankin-Selberg L-functions, Number Theory (Turku, 1999), (2001), 201–221.Search in Google Scholar
[24] Newton, J.—Thorne, J. A.: Symmetric power functoriality for holomorphic modular forms, Publ. Math. Inst. Hautes Études Sci. 134 (2021), 1–116.10.1007/s10240-021-00127-3Search in Google Scholar
[25] Perelli, A.: General L-functions, Ann. Mat. Pura Appl. 130 (1982), 287–306.10.1007/BF01761499Search in Google Scholar
[26] Rankin, R. A.: Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetical functions. I. The zeros of the function
[27] Selberg, A.: Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Natu. 43 (1940), 47–50.Search in Google Scholar
[28] 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 (2022), Art No. 19.10.1007/s40993-022-00319-8Search in Google Scholar
[29] Wei, L. L.—Lao, H. X.: The mean value of a hybrid arithmetic function associated to Fourier coefficients of cusp forms, Integers 19 (2019), Art. No. A44.Search in Google Scholar
[30] Xu, C. R.: General asymptotic formula of Fourier coefficients of cusp forms over sum of two squares, J. Number Theory 236 (2022), 214–229.10.1016/j.jnt.2021.07.017Search in Google Scholar
[31] Zhai, S.: Average behavior of Fourier coefficients of cusp forms over sum of two squares, J. Number Theory 133 (2013), 3862–3876.10.1016/j.jnt.2013.05.013Search in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Algebraic structures formalizing the logic of effect algebras incorporating time dimension
- Walks on tiled boards
- Composition on FLew-algebras
- On high power sums of a hybrid arithmetic function
- On indices of quintic number fields defined by x5 + ax + b
- Note on fundamental system of solutions to the differential equations (D2 − 2Dα + α2 ± β2) y = 0
- Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals
- A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications
- On a periodic problem for super-linear second-order ODEs
- Existence and Uniqueness of Solutions for Fractional Dynamic Equations with Impulse Effects
- Periodic Solutions for Conformable Non-autonomous Non-instantaneous Impulsive Differential Equations
- Self referred equations with an integral boundary condition
- Approximation by matrix means of double Vilenkin-Fourier series
- Maps on self-adjoint operators preserving some relations related to commutativity
- Chains in the Rudin-Frolík order
- Relative versions of first and second countability in hyperspaces
- Proportion estimation in multistage pair ranked set sampling
- The Marshall-Olkin Extended Gamma Lindley distribution: Properties, characterization and inference
Articles in the same Issue
- Algebraic structures formalizing the logic of effect algebras incorporating time dimension
- Walks on tiled boards
- Composition on FLew-algebras
- On high power sums of a hybrid arithmetic function
- On indices of quintic number fields defined by x5 + ax + b
- Note on fundamental system of solutions to the differential equations (D2 − 2Dα + α2 ± β2) y = 0
- Several sharp inequalities involving (hyperbolic) tangent, tanc, cosine, and their reciprocals
- A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications
- On a periodic problem for super-linear second-order ODEs
- Existence and Uniqueness of Solutions for Fractional Dynamic Equations with Impulse Effects
- Periodic Solutions for Conformable Non-autonomous Non-instantaneous Impulsive Differential Equations
- Self referred equations with an integral boundary condition
- Approximation by matrix means of double Vilenkin-Fourier series
- Maps on self-adjoint operators preserving some relations related to commutativity
- Chains in the Rudin-Frolík order
- Relative versions of first and second countability in hyperspaces
- Proportion estimation in multistage pair ranked set sampling
- The Marshall-Olkin Extended Gamma Lindley distribution: Properties, characterization and inference