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

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 ∞ [ a f ( n ) ψ ( n ) ∗ μ ( n ) ψ ′ ( n ) ] exp ( - n y ) , where a f ( n ) is the n th 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.