What question did this study set out to answer?

The aim is to explore the kinematic-geometric properties of the Riemann zeta function using center-of-mass dynamics.

April 28, 2026Open Access

Dirichlet Walks and Center-of-Mass Stability: A Kinematic-Geometric Approach to the Riemann Hypothesis

Key Points

The aim is to explore the kinematic-geometric properties of the Riemann zeta function using center-of-mass dynamics.
Developed a framework using finite Dirichlet sums as weighted Dirichlet walks.
Analyzed the center-of-mass evolution and its asymptotic helical structure through Euler--Maclaurin summation.
Decomposed wobble dynamics into symmetric and asymmetric components to evaluate critical-line behavior.
Identified that the absence of residual wobble is crucial for the localization of nontrivial zeros on the critical line Re(s)=1/2.
Showed that the asymmetric contribution to the wobble vanishes specifically on the critical line, linking it to geometric rigidity.

Abstract

This paper develops a kinematic-geometric framework for the Riemann zeta functionbased on the center-of-mass dynamics of finite Dirichlet sums. Interpreting the partial sums as a weighted Dirichlet walk, we study the accumulation of its normalized centerof mass and identify an asymptotic helical structure arising fromEuler--Maclaurin summation. Within this framework, zeta zeros are characterized by the disappearance ofleading residual wobble in the center-of-mass evolution. Analyzing the induced wobble dynamics, we decompose the forcing into symmetricand asymmetric components and show that the asymmetric contribution vanishes onlyon the critical line Re(s)=1/2. This yields a geometric rigidity mechanism linking the absence of residual wobbleto critical-line localization of nontrivial zeros.

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