On the Epstein zeta function
An elementary proof is given of the range of convergence of Epstein zeta functions \[ \zeta(T:s):= \sum_{0\neq g\in \mathbb{Z}^ n} (g^ t Tg)^{- s}, \] with \(T\) any real symmetric positive-definite \(n\times n\) matrix. In particular, a very simple proof of the following theorem is presented: the series converges absolutely for \(\text{Re} (s)> n/2\). The first proof of such theorem was due to \textit{G. Eisenstein} [J. Reine Angew. Math. 35, 137-184 (1847)] and some other proofs are also noted in the text. The new demonstration is very simple and is done by induction on \(n\). As a corollary it is seen in the paper that the above function admits a meromorphic continuation to the whole complex \(s\)-plane with a simple pole at \(s= n/2\), which is also a well-known result. Related brand new results on non-homogeneous Epstein zeta functions (in particular, an extension of the famous Chowla-Selberg formula) have been found by \textit{E. Elizalde} [J. Math. Phys. 35, 6100-6122 (1994), and an unpublished preprint].
scientific article; zbMATH DE number 604858
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