What question did this study set out to answer?

This research aims to establish a unified spectral framework linking prime Hamiltonians with the Riemann zeta function and random matrix theory.

April 25, 2026Open Access

Unified Spectral RG Framework for the Riemann Zeta Function and Random Matrix Theory

Key Points

This research aims to establish a unified spectral framework linking prime Hamiltonians with the Riemann zeta function and random matrix theory.
Constructed trace-class perturbations of diagonal prime operators.
Utilized Fredholm spectral determinants and RG flow of correlation kernels.
Analyzed fixed-point conditions to establish universality in correlation hierarchies.
Demonstrated that the sine-kernel determinantal process uniquely corresponds to the Gaussian Unitary Ensemble (GUE) under specific conditions.
Reformulated the Riemann Hypothesis as a spectral uniqueness criterion for RG-stable determinant classes.

Abstract

AbstractWe propose a unified spectral framework connecting logarithmic prime Hamiltonians, theRiemann zeta function, and random matrix theory. The construction is based on trace-classperturbations of diagonal prime operators, Fredholm spectral determinants, and renormalizationgroup (RG) flow of correlation kernels. We show that under positivity, translation invariance,and RG fixed-point conditions, the unique universal limit of the correlation hierarchy is thesine-kernel determinantal process corresponding to the Gaussian Unitary Ensemble (GUE).The Riemann Hypothesis is reformulated as a spectral uniqueness condition for the RG-stabledeterminant class.

Unified Spectral RG Framework for the Riemann Zeta Function and Random Matrix Theory

Key Points

Abstract

Cite This Study