An approach to F_1 via the theory of lambda rings

We model the field $F_1$ of one element as a lambda ring $\mathbf{Z}$ with the canonical lambda structure. We show that then we can calculate the Riemann zeta function of integers in two ways: the first, geometrical, as a zeta function of the affine line over $F_1$ and the second, categorical, as a zeta function of the category of modules over $F_1$.