An approach to F_1 via the theory of lambda rings

TL;DR

This paper explores a novel approach to understanding the Riemann zeta function at integers by modeling the hypothetical field F_1 as a lambda ring, connecting geometry and category theory.

Contribution

It introduces a lambda ring model for F_1 and demonstrates two methods to compute the zeta function of integers, linking geometry and category theory perspectives.

Findings

Zeta function of integers can be computed geometrically over F_1.

Categorical approach yields the same zeta function via modules over F_1.

Provides a new framework connecting lambda rings and number theory.

Abstract

We model the field $F_1$ of one element as a lambda ring $\bf Z$ 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 $F_1$.