An asymptotic expansion for a twisted Lambert series associated to a cusp form and the M\"{o}bius function: level aspect

TL;DR

This paper derives an asymptotic expansion for a twisted Lambert series linked to cusp forms and the Möbius function, extending previous work to higher level subgroups and character analogues.

Contribution

It generalizes the analysis of Lambert series associated with cusp forms to higher level subgroups and includes character analogues, broadening the scope of prior results.

Findings

Derived an asymptotic expansion for the Lambert series

Extended the analysis to higher level subgroups

Included character analogues in the series

Abstract

Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the M\"{o}bius function. In this paper, we investigate the Lambert series $\sum_{n=1}^{\infty}[a_f(n)\psi(n)*\mu(n)\psi'(n)]\exp(-ny),$ where $a_f(n)$ is the $n$th Fourier coefficient of a cusp form $f$ over any congruence subgroup, and $\psi$ and $ \psi'$ are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue.