Dirichlet series corresponding to Siegel's modular forms of degree n with level N
Let \(F(Z)=\sum_{T}a(T)e^{2\pi i trace(TZ)}\) be a Siegel modular form of degree n and weight k. A natural type of Dirichlet series attached to F is \[ L(F,s):=\sum_{\{T\}}a(T)/\epsilon (T) \det (T)^ s, \] where \(\{\) \(T\}\) runs through all positive definite semi-integral symmetric n by n matrices modulo the action of Gl(n,\({\mathbb{Z}})\) and \(\epsilon\) (T) denotes the number of units of T. Such series were formerly studied by Koecher and Maaß [see e.g. \textit{H. Maaß}'s book ``Siegel's modular forms and Dirichlet series'', Lect. Notes Math. 216 (1971; Zbl 0224.10028)], their residues however remained unknown, until the author (using Klingen type Eisenstein series) proved a residue formula for them [Math. Ann. 238, 157-173 (1978; Zbl 0373.10017)]. In the paper under review the author gives another proof of the residue formula, which is more complicated than the previous one, but it has two advantages: It avoids the use of Klingen-Eisenstein series and the restriction on the weight is weaker \((k>n-1\) instead of \(k>2n)\). The author's result covers the case of Siegel modular forms for congruence subgroups of type \(\Gamma^ n_ 0(N).\) By essentially the same method the author also computes the residues of Koecher-Epstein zeta-functions attached to positive definite quadratic forms. That residue formula was claimed in \textit{M. Koecher}'s dissertation [J. Reine Angew. Math. 192, 1-23 (1953; Zbl 0053.055)], but - as is well known - there is a gap in Koecher's proof.
residues of Koecher-Epstein zeta-functions
scientific article; zbMATH DE number 4175033
scientific article