Arrangements of hyperplanes II: Szenes formula and Eisenstein series
Abstract
Description
The aim of this article is to generalize in several variables some formulae for Eisenstein series in one variable. For example the formula $2ζ(2k) = (2π)^{2k} \frac{B_{2k}}{(2k)!} = Res_{z=0}(\frac{1}{z^{2k}(1-e^z)})$ 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.
revised version (introduction rewritten, references added, minor changes made), 28 pages, LaTEX2e
revised version (introduction rewritten, references added, minor changes made), 28 pages, LaTEX2e