  The renormalization of MZV was until now carried out by algebraic means. We
show that renormalization in general, of the multiple zeta functions in
particular, is more than mere convention. We show that simple calculus methods
allow us to compute the renormalized values of multiple zeta functions in any
dimension for arguments of the form (1,...,1), where the series do not
converge. These values happen to be the coefficients of the asymptotic
expansion of the inverse Gamma function. We focus on the geometric
interpretation of these values, and on the combinatorics their closed form
encodes, which happen to match the combinatorics of the generalized Euler
reflection formula discovered by Michael E. Hoffman, which in turn is a kind of
analogue of the Cayley-Hamilton theorem for matrices. By means of one single
limit formula, we define a function on the positive open half-line which takes
exactly the values of the Riemann zeta function, with the additional advantage
that it equals the Euler constant when the argument is 1.
