We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{ X}$ at any integer $n$ in terms of Weil-\'etale cohomology complexes. This extends work of Lichtenbaum \cite{ Lichtenbaum05} and Geisser \cite{ Geisser04b} for $\mathcal{ X}$ of characteristic $p$, of Lichtenbaum \cite{ li04} for $\mathcal{ X}=\mathrm{ Spec}(\mathcal{ O}_F)$ and $n=0$ where $F$ is a number field, and of the second author for arbitrary $\mathcal{ X}$ and $n=0$ \cite{ Morin14}. We show that our conjecture is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou \cite{ fpr91} if $\mathcal{ X}$ is smooth over $\mathrm{ Spec}(\mathcal{ O}_F)$, and hence that it holds in cases where the Tamagawa number conjecture is known.
Categories: math.NT math.AG