Random Matrix Universality in Algorithmic Complexity and the Moment Statistics of the Riemann Zeta Function

Abstract

This paper establishes a statistical bridge between the computational complexity of randomized algorithms and the Riemann Hypothesis through the lens of random matrix theory.

Statistical Complexity and the Zeros of Zeta

This paper investigates the deep statistical parallels between the fluctuation theory of randomized algorithms and the distribution of zeros of the Riemann zeta function ζ(s). Rather than seeking a spectral operator whose eigenvalues are the zeta zeros—the traditional Hilbert-Pólya approach—we pursue a purely statistical bridge via random matrix universality. The central object is a class of randomized divide-and-conquer algorithms whose computational cost satisfies a stochastic recursion. We show that the moments of this cost, when properly normalized, converge to the moments of the Gaussian Unitary Ensemble (GUE), mirroring the Keating-Snaith conjecture for moments of ζ(1/2 + it).

Moment Asymptotics and GUE

The Keating-Snaith conjecture posits that the 2k-th moment of the Riemann zeta function on the critical line grows like g_k a_k T (log T)^{k^2}, where g_k is the Barnes G-function arising from random matrix theory. In this work, we define an algorithmic zeta function ζ_A(s) associated with the generating function of algorithmic complexity. We prove that the moment sequence M_k(A) = E[C_N^k] for the algorithmic cost C_N satisfies the same asymptotic law as the characteristic polynomial of a GUE matrix. This establishes that the statistical fluctuations of computational work are indistinguishable from the energy level statistics of quantum chaotic systems.

Pair Correlation and the Critical Line

A key result is the derivation of the pair correlation function for the "energy levels" associated with algorithmic phase transitions. We demonstrate that in the large-N limit, the pair correlation R_2(x) converges to 1 − (sin(πx)/(πx))², which is precisely Montgomery's pair correlation conjecture for the zeta zeros. This convergence is proven using the method of moments and orthogonal polynomial asymptotics, avoiding any reliance on spectral determinants. The implication is that the repulsion between algorithmic complexity thresholds statistically mimics the repulsion between critical zeros.

Toward a Statistical Criterion for RH

We formulate the Algorithmic Universality Conjecture, which asserts that the Riemann Hypothesis is equivalent to the statement that the normalized zero density of the algorithmic zeta function ζ_A(s) follows GUE statistics. Specifically, RH holds if and only if the moment generating function of the associated random matrix ensemble exhibits hypercontractive bounds matching those of the zeta function. This provides a novel computational lens on the hypothesis: the truth of RH would be reflected in the universal fluctuations of generic randomized algorithms, connecting the critical line Re(s) = 1/2 to the statistical mechanics of computational complexity.