Abstract
This essay examines the structural resonance between the Floquet theory of periodically driven enzymatic reactions, developed by Watanabe, Ishiguro, and Oka, and the analytic theory of the Riemann zeta function.
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Overview
This essay explores a potential structural bridge between the Floquet theory of enzymatic reactions and the Riemann Hypothesis (RH). The source paper, "Floquet Driving of Enzymatic Reactions: Counting Statistics and Long-Time Currents" (arXiv:2607.17072v1), develops a formalism for calculating counting statistics in periodically driven chemical reactions. A key analytical result is the high-frequency (van Vleck) expansion of the effective Floquet generator, which explicitly involves the Hurwitz zeta function ζ(s, a).
The Hurwitz zeta generalizes the Riemann zeta function (where ζ(s) = ζ(s, 1)), and its appearance in a physical context often signals deep number-theoretic content. We investigate whether this shared mathematical object suggests a deeper analogy: perhaps the effective Floquet generator corresponds to a spectral determinant related to the zeta zeros, or the counting field χ corresponds to the complex variable s.
This analogy is proposed as speculative exploration only. We rate the structural correspondence as a suggestive metaphor rather than a formal analogy or isomorphism. While both domains employ the Hurwitz zeta in asymptotic expansions, the source domain lacks the critical structures that make RH difficult: the Euler product, the functional equation, and the infinite-dimensional unitary spectral problem. The enzymatic model is a finite-state, non-unitary classical stochastic process.
We adopt the computational experiment design angle, proposing algorithms to compute the spectral statistics of the Floquet operator and compare them to the Gaussian Unitary Ensemble (GUE) statistics observed in the Riemann zeros. The discriminating outcome would be whether the stochastic model exhibits spectral rigidity (supporting a deeper connection) or generic Poisson statistics (refuting it). We conclude with an honest assessment of why the analogy fails and what source structures would be needed to repair it.
This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.