Superfluous Non-Gaussianity in Langevin Dynamics and the Riemann Hypothesis: An Exploratory Negative

Abstract

We examine whether the structural patterns in Plyukhin's analysis of non-Gaussian thermal noise in generalized Langevin equations (GLE) suggest a bridge to the Riemann Hypothesis (RH) via probabilistic models and multiplicative chaos.

Overview

This essay assesses a potential structural bridge between stochastic thermodynamics and the Riemann Hypothesis (RH). The source paper by Plyukhin (arXiv:2601.16114v1) analyzes generalized Langevin equations (GLE) driven by non-Gaussian thermal noise. It proves that the Jarzynski equality—an exact constraint on the exponential average of work—holds universally only up to third order in the process duration τ; beyond this, the equality forces the noise to be Gaussian. This creates a "threshold" effect where low-order observables are insensitive to noise statistics, while higher-order moments enforce Gaussianity.

The suggested angle for this catalogue entry was probabilistic models and multiplicative chaos. The Riemann zeta function ζ(s) is modeled probabilistically by random multiplicative functions and, in the mesoscopic limit, by critical multiplicative chaos—Gaussian log-correlated fields. One might superficially hope that the "necessity of Gaussianity" in Plyukhin's fluctuation theorem could mirror the emergence of Gaussian universality in zeta's random models.

We rate this analogy as structurally broken. While both fields invoke Gaussianity as a critical assumption, the source structure involves additive noise in a continuous-time dynamical system constrained by a thermodynamic identity (Jarzynski). The RH side involves multiplicative randomness (Euler products) and complex-analytic constraints. There is no formal dictionary linking the Jarzynski expansion in τ to any known asymptotic in zeta theory, and the "superfluity" of non-Gaussianity for quadratic observables in the Langevin context has no arithmetic counterpart. This essay explains precisely where the correspondence collapses.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.