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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 finitetype 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. Weprove 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 $f$.

An arithmetic zeta function respecting multiplicities