ON EPSTEIN'S ZETA FUNCTION OF HUMBERT FORMS

Abstract

The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any complex number s such that ℜ s > n/2, and admits a meromorphic continuation to the complex plane, with a simple pole at s = n/2. The question as to which Γ, for a fixed s, minimizes ζ(Γ,s), has a long history, dating back to Sobolev's work on numerical integration, and subsequent papers by Delone and Ryshkov among others. This was also investigated more recently by Sarnak and Strombergsson. The present paper is concerned with similar questions for positive definite quadratic forms over number fields, also called Humbert forms. We define Epstein zeta functions in that context and study their meromorphic continuation and functional equation, this being known in principle but somewhat hard to find in the literature. Then, we give a general criterion for a Humbert form to be finally ζ extreme, which we apply to a family of examples in the last section.