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 $$ ζ(s;f,g)=\sum_{k_1=0}^\infty ... \sum_{k_p=0}^\infty g(k_1,...,k_p)f(k_1,...,k_p)^{-s}\ \,(\re(s)\gg0), $$ to special values of zeta integrals $$ Z(s;f,g)=\int_{x\in[0,\infty)^p} g(x)f(x)^{-s}\,dx \, \ (\re(s)\gg0).$$ We prove a simple relation between $ζ(-N;f,g)$ and $Z(-N;f_a,g_a)$, where for $a\in\C ^p,\ f_a(x)$ is the shifted polynomial $f_a(x)=f(a+x)$. By direct calculation we prove the product rule for zeta integrals at $s=0$, $\mathrm{degree}(fh)\cdot Z(0;fh,g)=\mathrm{degree}(f)\cdot Z(0;f,g)+\mathrm{degree}(h)\cdot Z(0;h,g),$ and deduce the corresponding rule for Dirichlet series at $s=0$, $\mathrm{degree}(fh)\cdotζ(0;fh,g)=\mathrm{degree}(f) \cdotζ(0;f,g)+\mathrm{degree}(h)\cdotζ(0;h,g).$ This last formula generalizes work of Shintani and Chen-Eie.

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