Distribution of the Riemann zeros represented by the Fermi gas

TL;DR

This paper explores the connection between the distribution of Riemann zeros and the correlation structure of a one-dimensional ideal Fermi gas, revealing a deep link with random matrix theory.

Contribution

It demonstrates that the correlation structure of a 1D Fermi gas matches that of eigenvalues of random unitary matrices, linking quantum gases to number theory.

Findings

Correlation kernel of 1D Fermi gas matches that of random unitary matrices.

The structure of Riemann zeros distribution is akin to eigenvalue distributions.

Implications for understanding the zeros of the Riemann zeta function are discussed.

Abstract

The multiparticle density matrices for degenerate, ideal Fermi gas system in any dimension are calculated. The results are expressed as a determinant form, in which a correlation kernel plays a vital role. Interestingly, the correlation structure of one-dimensional Fermi gas system is essentially equivalent to that observed for the eigenvalue distribution of random unitary matrices, and thus to that conjectured for the distribution of the non-trivial zeros of the Riemann zeta function. Implications of the present findings are discussed briefly.