Arrangements of hyperplanes II: Szenes formula and Eisenstein series

Details Arrangements of hyperplanes II: Szenes formula and Eisenstein series Arrangements of hyperplanes II: Szenes formula and Eisenstein series

1999-03-30

arxiv.org/abs/math/9903180v2

Mathematics

Differential Geometry

revised version (introduction rewritten, references added, minor changes made), 28 pages, LaTEX2e

Scientific paper

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.

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