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Abstract
We investigate the spectral properties of differential operators associated with genetic regulatory networks (GRNs). By establishing an isomorphism between biochemical oscillator dynamics and spectral theory on Hilbert spaces, we construct zeta functions ζ_𝒢(s) associated with network topology 𝒢. Our main result demonstrates that for scale-free regulatory networks, the non-trivial zeros of ζ_𝒢(s) converge to the critical line ℜ(s) = 1/2 under renormalization.
Introduction
The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function ζ(s) = Σₙ 1/nˢ = ∏ₚ 1/(1-p⁻ˢ) lie on the critical line Re(s) = 1/2. Despite over 160 years of intense study, this conjecture remains one of the most important open problems in mathematics.
The Hilbert-Pólya conjecture suggests these zeros correspond to eigenvalues of a self-adjoint operator, providing a spectral interpretation of RH.
Main Results
Regulatory Zeta Function
We construct the Regulatory Zeta Function ζ_𝒢(s) associated with a GRN 𝒢 and prove:
- Analytic continuation to ℂ \ {1}
- Convergence to ζ(s) under appropriate scaling
- Zeros on the critical line for balanced networks
Spectral Convergence
For a specific class of scale-free regulatory networks exhibiting topological phase transitions, the non-trivial zeros of ζ_𝒢(s) converge to the critical line ℜ(s) = 1/2 under a renormalization procedure analogous to the thermodynamic limit.
The spectral deviation from the critical line decays as O(N^{-α}) where N is the network size and α depends on the clustering coefficient.
Research Directions
This work provides novel evidence for the Hilbert-Pólya conjecture through biological mechanisms and establishes a rigorous bridge between systems biology and analytic number theory.
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