Remarks on zeta functions and K-theory over ${\mathbf F}_1$

Abstract

We show that the notion of zeta functions over the field of one element $\F_1$, as given in special cases by Soulé, extends naturally to all $\F_1$-schemes as defined by the author in an earlier paper. We further give two constructions of K-theory for affine schemes or $\F_1$-rings, we show that these coincide in the group case, but not in general.