Abstract
Hafner and Stopple proved a conjecture of Zagier relating to the asymptotic behaviour of the inverse Mellin transform of the symmetric square L-function associated with the Ramanujan tau function. In this paper, we prove a similar result for any cusp form over the full modular group.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berndt, B.C.: Ramanujan’s Notebooks Part V. Springer, New York (1998)
Chandrasekharan, K., Narasimhan, R.: Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. Math. 76(2), 93–136 (1962)
Dixit, A.: Character analogues of Ramanujan-type integrals involving the Riemann \(\Xi \) function. Pac. J. Math. 255(2), 317–348 (2012)
Dixit, A.: Analogues of the general theta transformation formula. Proc. R. Soc. Edinb. Sect. A 143, 371–399 (2013)
Dixit, A., Roy, A., Zaharescu, A.: Ramanujan–Hardy–Littlewood–Riesz type phenomena for Hecke forms. J. Math. Anal. Appl. 426, 594–611 (2015)
Dixit, A., Roy, A., Zaharescu, A.: Riesz-type criteria and theta transformation analogues. J. Number Theory 160, 385–408 (2016)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Based, in Part, on Notes Left by Harry Bateman, vols. I–III. McGraw-Hill, New York (1953)
Hardy, G.H., Littlewood, J.E.: Contributions to the theory of Riemann zeta-function and the theory of the distribution of primes. Acta Math. 41, 119–196 (1916)
Hafner, J., Stopple, J.: A heat kernel associated to Ramanujan’s tau function. Ramanujan J. 4, 123–128 (2000)
Kanemitsu, S., Tsukada, H.: Contributions to the Theory of Zeta-Functions–Modular Relation Supremacy. World Scientific, London (2014)
Rankin, R.A.: Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions, I, II. Proc. Camb. Philos. Soc. 35, 351–372 (1939)
Roy, A., Zaharescu, A., and Zaki, M.: Some identities involving convolutions of Dirichlet characters and the Möbius function. Proc. Math. Sci. (to appear)
Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. Lond. Math. Soc., Ser. 31(3), 79–98 (1975)
Titchmarsh, E.C.: The Theory of the Riemann Zeta Function. Clarendon Press, Oxford (1986)
Zagier, D.: The Rankin–Selberg method for automorphic functions which are not of rapid decay. J. Fac. Sci. Univ. Tokyo IA Math. 28, 415–437 (1981)
Acknowledgments
The authors are indebted to the referee for valuable comments which helped in improving the manuscript immensely.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Chakraborty, K., Kanemitsu, S. & Maji, B. Modular-type relations associated to the Rankin–Selberg L-function. Ramanujan J 42, 285–299 (2017). https://doi.org/10.1007/s11139-015-9759-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-015-9759-8