A theorem on zeta functions associated with polynomials

Let beta = (beta(1),...,beta(r)) be an r-tuple of non-negative integers and P-j(X) (j = 1, 2,...,n) be polynomials in R [X-1,...,X-r] such that P-j(n) > 0 for all n epsilon N-r and the series [GRAPHICS] is absolutely convergent for Re s > sigma(j) > 0. We consider the zeta functions [GRAPHICS] All these zeta functions Z(Pi(j)(n) = 1 P-j, beta, s) and Z(P-j, beta, s) (j = 1, 2,...,n) are analytic functions of s when Re s is sufficiently large and they have meromorphic analytic continuations in the whole complex plane. In this paper we shall prove that [GRAPHICS] As an immediate application, we use it to evaluate the special values of zeta functions associated with products of linear forms as considered by Shintani and the first author.