In this paper, we study the arithmetic zeta function $$\mathscr{ Z}_{ \mathcal{ X}}(s) = \prod_p \prod_{ \substack{ x \in \mathcal{ X}_p \\ \text{ closed}}} \Big( \frac{ 1}{ 1-|\kappa(x)|^{ -s}} \Big)^{ \mathfrak{ m}_{ p}(x)}$$ associated to a scheme $\mathcal{ X}$ of finite type over $\mathbb{ Z}$, where $\kappa(x)$ denotes the residue field and $\mathfrak{ m}_{ p}(x)$ the multiplicity of $x$ in $\mathcal{ X}_p$. If $\mathcal{ X}$ is defined over a finite field, then $\mathscr{ Z}_{ \mathcal{ X}}$ appears naturally in the context of point counting with multiplicities. We prove that $\mathscr{ Z}_{ \mathcal{ X}}$ admits a meromorphic continuation to $\{ s \in \mathbb{ C} \colon \mathrm{ Re}(s) > \mathrm{ dim}(\mathcal{ X})-1/2\}$ and determine the order of its pole at $s = \mathrm{ dim}(\mathcal{ X})$. Finally, we relate $\mathscr{ Z}_{ \mathcal{ X}}$ to a zeta function $\zeta_f$ encoding the residual factorization patterns of a polynomial $f$.
Categories: math.NT math.AG