SPECIAL VALUES OF DIRICHLET SERIES AND ZETA INTEGRALS

For f and g polynomials in p variables, we relate the special value at a non-positive integer s = -N, obtained by analytic continuation of the Dirichlet series zeta(s; f, g) = Sigma(infinity)(k1=0)...Sigma(infinity)(kp=0) g(k(1),..., k(p)) f (k(1),..., k(p))(-s) (Re(s) >> 0), to special values of zeta integrals Z(s; f, g) = integral(x is an element of[0,infinity)p) g(x) f(x)(-s) dx (Re(s) >> 0). We prove a simple relation between zeta(-N; f, g) and Z(-N; f(a), g(a)), where for a is an element of C-p, f(a)(x) is the shifted polynomial f(a)(x) = f(a + x). By direct calculation we prove the product rule for zeta integrals at s = 0, degree(fh) . Z(0; fh, g) = degree(f) . Z(0; f, g) + degree(h) . Z(0; h, g), and deduce the corresponding rule for Dirichlet series at s = 0, degree(fh) . zeta(0; fh, g) = degree(f) . zeta(0; f, g)+ degree(h) . zeta(0; h, g). This last formula generalizes work of Shintani and Chen-Eie.