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

TL;DR

This paper derives an asymptotic expansion for a Lambert series linked to the symmetric square $L$-function of a Hecke eigenform, extending previous results related to the Riemann zeta function zeros.

Contribution

It generalizes the asymptotic expansion of Lambert series associated with symmetric square $L$-functions for Hecke eigenforms over the full modular group.

Findings

Established an asymptotic expansion as y approaches 0+

Extended previous results to a broader class of modular forms

Connected the expansion to the zeros of the Riemann zeta 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 non-trivial zeros of the Riemann zeta function $\zeta(s)$. Later, Chakraborty, Kanemitsu and the second author extended this phenomenon for any Hecke eigenform over the full modular group. In this paper, we study an asymptotic expansion of the Lambert series \begin{equation*} y^k \sum_{n=1}^\infty \lambda_{f}( n^2 ) \exp (- ny), \quad \textrm{as}\,\, y \rightarrow 0^{+}, \end{equation*} where $\lambda_f(n)$ is the $n$th Fourier coefficient of a Hecke eigen form $f(z)$ of weight $k$ over the full modular group.