An Asymptotic Expansion for a Twisted Lambert Series Associated to a Cusp Form and the Möbius Function: Level Aspect

Abstract

Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the Möbius function. In this paper, we investigate the Lambert series ∑n=1∞[af(n)ψ(n)∗μ(n)ψ′(n)]exp(-ny), where af(n) is the nth Fourier coefficient of a cusp form f over any congruence subgroup, and ψ and ψ′ are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.