Regularized Products over arithmetic Schemes
Mounir Hajli
TL;DR
This work develops a super-regularization framework for infinite products over arithmetic schemes by extending zeta-regularization with a two-variable zeta function and residue calculus. It defines zeta-type sums $\mathcal{P}_K(s)$ and $\mathcal{P}_{m,a}(s)$ to regularize products of prime norms in both number fields and arithmetic progressions, and derives explicit closed-form expressions in number-field and function-field settings. The main results express regularized prime-product invariants in terms of zeta-derivative data and arithmetic invariants (e.g., $\zeta_K^{(r_1+r_2)}(0)$, $h_K$, $R_K$, $w_K$) and provide concrete finite-field examples showing irrationality of the invariants and the infinitude of closed points. This framework attaches transcendental arithmetic invariants to schemes and suggests a broad program linking zeta-regularization with zeta/L-functions in arithmetic geometry.
Abstract
In this paper, we study the closed points of arithmetic schemes. We accomplish this by showing that the product of the cardinals of residue fields of closed points in an arithmetic scheme can be regularized. This regularization yields a new arithmetic invariant attached to the scheme. We compute it explicitly in several cases and find that it is always a transcendental number. This result provides a proof that the set of closed points is infinite. Our main tool is a regularization technique, which generalizes the zeta-regularization method introduced by Muñoz and Pérez.