Nested sums of symbols and renormalised multiple zeta functions
Abstract
We define discrete nested sums over integer points for symbols on the real line, which obey stuffle relations whenever they converge. They relate to Chen integrals of symbols via the Euler-MacLaurin formula. Using a suitable holomorphic regularisation followed by a Birkhoff factorisation, we define renormalised nested sums of symbols which also satisfy stuffle relations. For appropriate symbols they give rise to renormalised multiple zeta functions which satisfy stuffle relations at all arguments. The Hurwitz multiple zeta functions fit into the framework as well. We show the rationality of multiple zeta values at nonpositive integer arguments, and a higher-dimensional analog is also investigated.
Keywords
Cite
@article{arxiv.math/0702135,
title = {Nested sums of symbols and renormalised multiple zeta functions},
author = {Dominique Manchon and Sylvie Paycha},
journal= {arXiv preprint arXiv:math/0702135},
year = {2009}
}
Comments
Two major changes : improved treatment of the Hurwitz multiple zeta functions, and more conceptual (and shorter) approach of the multidimensional case