Abstract
This essay attempts to map the infinite hierarchy of coupled reaction-diffusion equations studied by Madeira, Ortgiese, and Penington—governing the spatial Muller's ratchet—onto the analytic growth theory of the Riemann zeta function, specifically moment estimates and the Lindelöf Hypothesis.
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Overview
This exploratory essay examines the deterministic spatial Muller's ratchet model introduced by Madeira, Ortgiese, and Penington (arXiv:2607.12864v1). The model consists of an infinite system of reaction-diffusion equations for population densities uk(t,x), where k counts deleterious mutations. The authors derive quantitative bounds on the ratios uk/u0 using a Feynman-Kac representation, establishing that deleterious mutations cannot "surf" the expansion wave.
We investigated whether this hierarchical bound structure could analogize to growth estimates for the Riemann zeta function ζ(s) on the critical line Re(s) = 1/2, such as the Lindelöf Hypothesis or moment asymptotics. The proposed dictionary mapped the mutation count k to the moment order and the spatial front to the critical line.
The analogy is rated a failed formal analogy. While both domains involve recursive inequalities controlling higher-order quantities, the specific mathematical structures diverge critically: the PDE hierarchy possesses a spatial diffusion operator and a Poisson stationary profile in k-space, whereas zeta moments lack a corresponding hierarchical convolution structure and exhibit no Poisson asymptotics. The essay concludes with an analysis of the obstructions and suggests that a viable bridge would require a source model with a spectral parameter naturally indexed by integers, rather than a spatial biological model.
This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.