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A Formal Group Perspective on the Riemann Zeta Function

Takao Inoué

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 $θ(x)-x$. 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 $\mathbb{F}_1$.

A Formal Group Perspective on the Riemann Zeta 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 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 . 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 .
Paper Structure (57 sections, 17 theorems, 103 equations)

This paper contains 57 sections, 17 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Formal Euler Products and Logarithmic Linearization
  3. One-Variable Formalization
  4. Formal Completion and Evenization
  5. Finite Cutoff Model
  6. Normalization and Gaussian Limit
  7. Fluctuation Decomposition
  8. Formal Group Preliminaries
  9. Formal Group Laws and Strict Isomorphisms
  10. Formal Group Logarithms
  11. The Multiplicative Formal Group
  12. Euler Factors as Formal Group Elements
  13. Euler Products as Formal Group Elements
  14. Formal Group-Valued Euler Products
  15. Logarithmic Linearization
  16. ...and 42 more sections

Key Result

Lemma 2.2

The logarithm of $\zeta^{\mathrm{for}}$ is given by

Theorems & Definitions (62)

  • Definition 2.1: Formal zeta element
  • Lemma 2.2: Formal logarithm
  • proof
  • Remark 2.3
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Definition 4.1: Odd and even parts
  • Definition 4.2: Formal completed zeta
  • ...and 52 more