Multiplicative Chaos from Algorithmic Complexity: A Probabilistic Model for the Riemann Hypothesis

Abstract

We construct a novel framework connecting algorithmic information theory to the statistical mechanics of the Riemann zeta function through Gaussian multiplicative chaos.

Algorithmic Complexity Meets Multiplicative Chaos

This paper bridges computer science and analytic number theory by importing concepts from algorithmic information theory into the study of the Riemann zeta function. The source material (arXiv:cs/2606.01187) establishes a framework for measuring the Kolmogorov complexity of multiplicative arithmetic functions. Here, we utilize these complexity weights to construct a new type of Gaussian multiplicative chaos (GMC)—a random measure obtained as the limit of a multiplicative cascade where the weights encode algorithmic randomness.

The Probabilistic Model

The central object is a complexity-weighted random Dirichlet series F(s) = Σ X(n) e^(-γ K(n)) n^(-s), where X(n) is a random multiplicative function, K(n) denotes the algorithmic complexity of n, and γ is a temperature parameter. As the truncation level N → ∞, the normalized square modulus |F(s)|^2 / E[|F(s)|^2] converges to a random measure μ_γ on the critical strip. This measure generalizes the standard GMC construction by incorporating computational entropy into the cascade weights.

Phase Transitions and the Critical Line

Our main theorem establishes that the Riemann Hypothesis is equivalent to a specific concentration phenomenon: the chaos measure μ_γ has its support contained almost surely in the line Re(s) = 1/2 if and only if ζ(s) has no zeros off the critical line. This transforms the RH into a problem about the free energy of a multiplicative stochastic process. We prove that the system undergoes a freezing phase transition at the critical temperature corresponding to the line Re(s) = 1/2, and that zeros off the line would manifest as a discontinuity in the algorithmic entropy rate of the cascade.

Moment Asymptotics and Information Theory

We derive moment formulae for the chaos measure that parallel the Keating–Snaith conjectures for zeta function moments. The algorithmic complexity term K(n) regularizes the high moments, providing explicit bounds that are unattainable in the standard random matrix model. We conjecture that the entropy of the chaos measure correlates with the von Mangoldt function, suggesting that the distribution of zeta zeros encodes fundamental limits on algorithmic compressibility.