Special values of Dirichlet series and zeta integrals

Abstract

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 ζ(s;f, g) = ∑ k1 = 0 ∞⋯∑ 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) = ∫ x∈[0, ∞)p g(x)f(x) -s dx (Re(s) ≫ 0). We prove a simple relation between ζ(-N;f,g) and Z(-N;f a, g a), where for a ∈ 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)·ζ(0;fh, g) = degree(f)·ζ(0;f, g)+degree(h)·ζ(0;h, g). This last formula generalizes work of Shintani and Chen-Eie. © 2012 World Scientific Publishing Company.