An asymptotic expansion of a Lambert series associated to cusp forms

Abstract

Zagier’s conjecture on the asymptotic expansion of the Lambert series ∑n=1∞τ2(n)exp(−nz), where τ(n) is the Ramanujan’s tau function, was proved by Hafner and Stopple. Recently, Chakraborty, Kanemitsu and Maji have extended this result to any cusp forms over the full modular group. The goal of this paper is to extend the asymptotic behavior to cusp forms over any congruence subgroup of the full modular group.