A Formal Group Perspective on the Riemann Zeta Function

TL;DR

This paper introduces a formal group framework for the Riemann zeta function, providing an algebraic linearization of its Euler product and revealing a Gaussian structure arising from the multiplicative formal group.

Contribution

It develops a novel algebraic formal group approach to analyze the Riemann zeta function, emphasizing structural and geometric aspects over spectral methods.

Findings

Gaussian leading term in the logarithmic expansion

Higher cumulants encode arithmetic deviations

Decomposition governed by Chebyshev error function

Abstract

We develop a formal group--theoretic framework for the Riemann zeta function by treating its Euler product as an element of the multiplicative formal group $\widehat{\mathbb{G}}_m$ and its logarithm as the associated formal group logarithm. This provides a purely algebraic linearization of the multiplicative prime-wise structure of the Euler product. Starting from a finite cutoff model, we introduce a formal completion via evenization and a natural normalization procedure. The resulting logarithmic expansion exhibits a Gaussian leading term, while higher-order terms form a hierarchy of cumulants. We show that this Gaussian structure is not probabilistic in origin, but arises from the infinitesimal quadratic geometry of the multiplicative formal group after linearization. The higher cumulants encode arithmetic deviations and admit a decomposition governed by weighted integrals of the Chebyshev error function $\theta(x)-x$. A systematic formal group axiomatization is then provided, placing the cutoff model and cumulant hierarchy within a coherent algebraic framework. The approach is deliberately non-spectral, focusing on a structural reorganization of the Euler product rather than on operator-theoretic realizations of zeros. We briefly indicate possible connections with absolute arithmetic and geometry over $\mathbb{F}_1$.