Relations between values at T -tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables

Abstract

We give a new and very concise proof of the existence of a holomorphic continuation for a large class of twisted multivariable zeta functions. To do this, we use a simple method of “decalage” that avoids using an integral representation of the zeta function. This allows us to derive explicit recurrence relations between the values at $T$–tuples of negative integers. This also extends some earlier results of several authors where the underlying polynomials were products of linear forms.