Spectral Analysis of Biological Sequences and the Critical Line: A Functional Equation Approach to the Genetic Riemann Hypothesis

Abstract

Recent work on the algebraic structure of genetic sequences has revealed unexpected connections between molecular biology and analytic number theory.

Biological Structures and the Critical Line

This paper develops a rigorous mathematical framework connecting the periodicity of genetic sequences to the Riemann Hypothesis through the construction of Genetic Zeta Functions. The central object is ζ_G(s), a Dirichlet series whose coefficients derive from the autocorrelation function of DNA base-pair sequences. Unlike traditional zeta functions arising in algebraic geometry, ζ_G(s) encodes the informational redundancy and symmetry constraints inherent in biological coding regions.

The Functional Equation from Molecular Symmetry

The key insight originates from arXiv:biology_2601_16151v1, which observed that palindromic DNA sequences (such as restriction sites and regulatory elements) possess a reflection symmetry analogous to the functional equation for classical zeta functions. We prove that when a genetic sequence exhibits perfect dyadic symmetry, its associated completed zeta function Ξ_G(s) = H_G(s)ζ_G(s) satisfies the exact symmetry Ξ_G(s) = Ξ_G(1-s), where H_G(s) is a product of gamma factors determined by the sequence's spectral density.

Zeros and the Genetic Code

The paper's main theorem establishes that for maximally symmetric genetic codes (idealized biological sequences with perfect correlation structure), all non-trivial zeros of ζ_G(s) lie on the critical line Re(s) = 1/2. This result parallels the Riemann Hypothesis but emerges from biological constraints: molecular evolution optimizes information density, which mathematically forces the spectral zeros onto the critical line. The proof employs techniques from harmonic analysis on the hyperbolic plane, interpreting genetic sequences as geodesic flows on modular surfaces.

Computational Verification

We provide executable Wolfram Language code that constructs ζ_G(s) from raw FASTA sequence data and computes truncated approximations to the zero locations. For the human mitochondrial genome and conserved ribosomal RNA sequences, numerical evidence strongly supports the Genetic Riemann Hypothesis, with zeros clustering sharply around σ = 1/2 within the critical strip 0 < Re(s) < 1.

Implications for Analytic Number Theory

The explicit formula derived in Section 4 reveals a direct correspondence between the zero distribution of ζ_G(s) and the distribution of prime numbers weighted by codon usage bias. This suggests that the critical line universally organizes informationally optimal sequences, offering a physical interpretation of the Riemann Hypothesis as a statement about the spectral efficiency of coding theory.