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 ${\sum }_{n=1}^{\infty }[a_f(n)\psi (n)\ast \mu (n){\psi }^{\prime }(n)]\exp (-ny),$ where $a_f(n)$ is the $n$th Fourier coefficient of a cusp form $f$ over any congruence subgroup, and $\psi$ and ${\psi }^{\prime }$ are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue.

15 pages, comments are welcome!