An asymptotic expansion for a Lambert series associated with Siegel cusp forms (Q6596354)

The purpose of this paper is to study certain Lambert series. Let \(Z= \left( \begin{smallmatrix} \tau & z \\ z & \tau' \end{smallmatrix} \right)\in \mathcal{H}_2\), where \(\tau, \tau' \in \mathcal{H}, z \in \mathbb{C}\). Let \(F_1\) and \(F_2\) be two Siegel cusp forms of weight \(k\) and degree \(2\) with the Fourier-Jacobi expansion given for by \N\[\NF_1(Z) =\sum_{n=1}^\infty \phi_n(\tau,z) \exp(2\pi in\tau'),\N\]\Nand \N\[\NF_2(Z) =\sum_{n=1}^\infty \psi_n(\tau,z) \exp(2\pi in\tau'). \N\]\NLet define an arithmetic function \(a_{F_1,F_2} (n)\) that satisfies the following generating function: \N\[\N\sum_{n=1}^\infty \frac{a_{F_1,F_2} (n)}{n^s}=\frac{D_{F_1, F_2}(s)}{\zeta(2s+1-2k)}, \N\]\Nwhere \N\[\ND_{F_1, F_2}(s)=\zeta(2s-2k+4)\sum_{n=1}^\infty \frac{\langle\phi_n, \psi_n\rangle}{n^s},\N\]\Nand \(\langle\phi_n, \psi_n\rangle\) denotes the Petersson scalar product of Jacobi cusp forms \(\phi_n\) and \(\psi_n\).\N\NUnder the assumption of the simplicity hypothesis of the non-trivial zeros of \(\zeta(s)\), the authors express in terms of the non-trivial zeros of the Riemann zeta function the following Lambert series \N\[\N\sum_{n=1}^\infty \langle\phi_n, \psi_n\rangle \exp(-4\pi n \alpha).\N\]\NThis estimate generalizes a prior result of \textit{J. Hafner} and \textit{J. Stopple} [Ramanujan J. 4, No. 2, 123-128 (2000; Zbl 0987.11055)].