An asymptotic expansion for a Lambert series associated to Siegel cusp forms of degree $n$

Abstract

Utilizing inverse Mellin transform of the symmetric square $L$-function attached to Ramanujan tau function, Hafner and Stopple proved a conjecture of Zagier, which states that the constant term of the automorphic function $y^{12}|\Delta (z){|}^2$ i.e., the Lambert series $y^{12}{\sum }_{n=1}^{\infty }\tau (n{)}^2{e}^{-4\pi ny}$ can be expressed in terms of the non-trivial zeros of the Riemann zeta function. This study examines certain Lambert series associated to Siegel cusp forms of degree $n$ twisted by a character $\chi$ and observes a similar phenomenon.

15 pages, comments are welcome! arXiv admin note: text overlap with arXiv:2305.07412