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Arrangement of hyperplanes. II: The Szenes formula and Eisenstein series. (English) Zbl 0968.32016

Motivated by the work of E. Witten on the symplectic volume of the space of homomorphisms of the fundamental group of a Riemann surface, D. Zagier introduced a series and gave a residue formula for it, by considering a sequence of linear forms in complex variables with integral coefficients for some numbers related with Bernoulli numbers.
The authors of this paper define the Eisenstein series by introducing the oscillating factor and give an explicit residue formula for it. By using the Eisenstein series, another proof of the Szenes residue formula is given.

MSC:

32S22 Relations with arrangements of hyperplanes
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References:

[1] M. Brion and M. Vergne, Arrangement of hyperplanes, I: Rational functions and Jeffrey-Kirwan residue , Ann. Sci. École Norm. Sup. (4) 32 (1999), 715–741. · Zbl 0945.32003 · doi:10.1016/S0012-9593(01)80005-7
[2] L. Jeffrey and F. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface , Ann. of Math. (2) 148 (1998), 109–196. JSTOR: · Zbl 0949.14021 · doi:10.2307/120993
[3] A. Szenes, Iterated residues and multiple Bernoulli polynomials , Internat. Math. Res. Notices 1998 , 937–956. · Zbl 0968.11015 · doi:10.1155/S1073792898000567
[4] E. Witten, On quantum gauge theories in two dimensions , Comm. Math. Phys. 141 (1991), 153–209. · Zbl 0762.53063 · doi:10.1007/BF02100009
[5] D. Zagier, “Values of zeta functions and their applications” in First European Congress of Mathematics (Paris, 1992), Vol. II, Progr. Math. 120 , Birkhäuser, Basel, 1994, 497–512. · Zbl 0822.11001
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