An asymptotic expansion for a Lambert series associated with Siegel cusp forms

Abstract

In 2000, Hafner and Stopple proved a conjecture of Zagier which states that the constant term of the automorphic function \(|\Delta (x+iy)|^2\), i.e., the Lambert series \(\sum _{n=1}^\infty \tau (n)^2 e^{-4 \pi n y}\), can be expressed in terms of the non-trivial zeros of the Riemann zeta function. In this article, we study an asymptotic expansion of a generalized version of the aforementioned Lambert series associated with Siegel cusp forms.