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The renormalization of MZV was until now carried out by algebraic means. Weshow that renormalization in general, of the multiple zeta functions inparticular, is more than mere convention. We show that simple calculus methodsallow us to compute the renormalized values of multiple zeta functions in anydimension for arguments of the form (1,...,1), where the series do notconverge. These values happen to be the coefficients of the asymptoticexpansion of the inverse Gamma function. We focus on the geometricinterpretation of these values, and on the combinatorics their closed formencodes, which happen to match the combinatorics of the generalized Eulerreflection formula discovered by Michael E. Hoffman, which in turn is a kind ofanalogue of the Cayley-Hamilton theorem for matrices. By means of one singlelimit formula, we define a function on the positive open half-line which takesexactly the values of the Riemann zeta function, with the additional advantagethat it equals the Euler constant when the argument is 1.

Analytic renormalization of multiple zeta functions. Geometry and combinatorics of the generalized Euler reflection formula for MZV