Special values of Dirichlet series and zeta integrals

2011/05/13 by Eduardo Friedman, Friedman, Eduardo, Aldo Pereira +1 · 1 citation
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1105.2603

arxiv created 2011/05/13 · arxiv updated 2011/05/16

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(k₁,...,k_p)f(k₁,...,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.

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• Polynomials associated to Lie algebras

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