Arrangements of hyperplanes II: Szenes formula and Eisenstein series

Abstract

The aim of this article is to generalize in several variables some formulae for Eisenstein series in one variable. For example the formula $2\zeta(2k) = (2\pi)^{2k} \frac{B_{2k}}{(2k)!} = Res_{z=0}(\frac{1}{z^{2k}(1-e^z)})$ for the values of zeta functions at even integers in functions of Bernoulli numbers. A. Szenes proved in several variables a similar residue formula for the values of the zeta function introduced by Witten. We introduce some Eisenstein series by averaging over a lattice rational functions with poles in an arrangement of hyperplanes. We give another proof of Szenes residue formula by relating it to the constant term of these Eisenstein series.