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

TL;DR

This paper investigates Lambert series linked to Siegel cusp forms of degree n, revealing asymptotic expansions related to the zeros of the Riemann zeta function, extending prior results on automorphic functions.

Contribution

It extends the analysis of Lambert series associated with automorphic forms to Siegel cusp forms of arbitrary degree n, deriving asymptotic expansions involving zeta zeros.

Findings

Derived an asymptotic expansion for Lambert series of Siegel cusp forms

Connected Lambert series behavior to non-trivial zeros of the Riemann zeta function

Extended Zagier's conjecture to higher degree automorphic forms

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 $y^{12}\sum_{n=1}^\infty \tau(n)^2 e^{-4 \pi n y}$ 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.