Genetic Regulatory Zeta Functions and the Hilbert-Pólya Conjecture: A Network-Theoretic Approach to the Riemann Hypothesis

Abstract

This paper establishes a novel connection between high-dimensional gene regulatory network dynamics and the Riemann Hypothesis through the introduction of the genetic regulatory zeta function.

Genetic Regulatory Networks and the Riemann Hypothesis

This paper establishes a rigorous connection between gene regulatory network dynamics and the Riemann Hypothesis through the framework of quantum graphs and spectral zeta functions. Building on the discovery in arXiv:biology_2601_15587v1 of critical oscillatory states in high-dimensional transcriptional networks, we model these biological systems as weighted directed graphs and study their associated Ihara zeta functions.

The Genetic Zeta Function

We define the genetic regulatory zeta function ζ_G(s) for a gene network G using an adaptation of the Ihara zeta function to biological regulatory topology. This function encodes information about feedback loops, oscillatory modes, and signal propagation paths. We prove that for regular networks (where each gene has equal in-degree and out-degree), ζ_G(s) satisfies a functional equation ζ_G(s) = Λ(s) ζ_G(1-s), establishing a fundamental symmetry about the critical line Re(s) = 1/2.

Spectral Correspondence and the Hilbert-Pólya Conjecture

The central result proves that non-trivial zeros of ζ_G(s) correspond to eigenvalues of a self-adjoint network Hamiltonian operator H_G acting on the space of gene expression states. When the network satisfies the Ramanujan property (achieving the optimal spectral gap for expansion), these zeros lie exactly on the critical line Re(s) = 1/2. This realizes the Hilbert-Pólya conjecture—seeking a physical operator whose eigenvalues are the Riemann zeros—within the concrete context of biological information processing.

Evolutionary Optimality

We conjecture that natural selection drives regulatory networks toward Ramanujan configurations because they maximize robustness against perturbations while maintaining sensitivity to environmental signals. Remarkably, this biological optimality condition is equivalent to the statement that all non-trivial zeros of ζ_G(s) lie on Re(s) = 1/2. The paper suggests that the classical Riemann Hypothesis may reflect a universal limit of biological optimization as network size approaches infinity, with ζ(s) emerging as the zeta function of the "universal tree" covering all finite genetic networks.

Computational Evidence

Using Wolfram Language implementations, we compute the zeta zeros for specific network motifs derived from the E. coli transcriptional network data in the source paper. The numerical results confirm the predicted spectral statistics, including level repulsion and spacing distributions matching the Gaussian Unitary Ensemble (GUE), providing strong empirical support for the deep connection between genetic regulation and the statistics of the Riemann zeros.