Random matrices and Riemann hypothesis

TL;DR

This paper explores the intriguing link between eigenvalue spacings of random matrices and the zeros of the Riemann zeta function through advanced algebraic and geometric frameworks rooted in the Langlands program.

Contribution

It introduces a novel approach connecting random matrix theory and the Riemann hypothesis using shifted quantized conjugacy classes within bilinear algebraic semigroups.

Findings

Identifies a symmetry via the differential bilinear Galois semigroup.

Proposes a geometric dynamical framework for the Riemann hypothesis.

Links eigenvalue spacings to zeros of the zeta function through algebraic structures.

Abstract

The curious connection between the spacings of the eigenvalues of random matrices and the corresponding spacings of the non trivial zeros of the Riemann zeta function is analyzed on the basis of the geometric dynamical global program of Langlands whose fundamental structures are shifted quantized conjugacy class representatives of bilinear algebraic semigroups.The considered symmetry behind this phenomenology is the differential bilinear Galois semigroup shifting the product,right by left,of automorphism semigroups of cofunctions and functions on compact transcendental quanta.