On zeta functions associated with polynomials

Abstract

We give direct proofs of meromorphic continuality on the whole complex plane of certain zeta functions ZP, Q (s) and Z (P/Q, s) associated with a pair of polynomials P, Q. We calculate ZP, Q (−q) (q a non-negative integer) and give explicit formulas for the residues of Z (P/Q, s) at poles.