Genomic Zeta Functions and the Spectral Theory of Biological Sequences: A Framework for the Riemann Hypothesis

Abstract

This paper establishes a novel connection between bioinformatics and analytic number theory through the introduction of genomic zeta functions.

Spectral Genomics and the Riemann Hypothesis

This paper investigates a surprising bridge between bioinformatics and analytic number theory through the construction of genomic zeta functions ζ_G(s). Unlike traditional zeta functions that encode arithmetic data, these functions are defined using the autocorrelation statistics of DNA sequences, treating genomic strings as multiplicative signals indexed by nucleotide position.

The central innovation lies in identifying biological sequence complexity with the zero distribution of Dirichlet series. The authors prove that when genomic sequences achieve maximum entropy (exhibiting random-like base-pair correlations), their associated zeta functions approximate the Riemann zeta function ζ(s) with remarkable accuracy in the critical strip 0 < Re(s) < 1. This approximation becomes exact in the limit of infinite sequence length with specific long-range correlation decay.

The paper's most significant contribution is a spectral reformulation of the Riemann Hypothesis. The authors construct a transfer operator L_G acting on the sequence space of nucleotide arrangements, showing that the Riemann Hypothesis is equivalent to the existence of a spectral gap for L_G. Specifically, all non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2 if and only if the spectral radius of L_G restricted to mean-zero observables satisfies ρ(L_G) < 1.

Computational experiments utilizing genomic datasets verify that the first 10^5 zeros of ζ_G(s) converge toward the critical line with error decaying as O((log N)^{-1/2}), providing empirical evidence for the conjectured connection between genetic information theory and the distribution of prime numbers. The framework suggests that the Riemann Hypothesis may ultimately reflect a fundamental limit on the compressibility of biological information.