An asymptotic expansion for a Lambert series associated to the symmetric square L -function

Abstract

Hafner and Stopple proved a conjecture of Zagier that the inverse Mellin transform of the symmetric square L-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function ζ(s). Later, Chakraborty et al. extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series ykΣ n=1∞λ f(n2)exp(-ny),as y → 0+, where λf(n) is the nth Fourier coefficient of a Hecke eigenform f(z) of weight k over the full modular group. © 2023 World Scientific Publishing Company.