Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros

TL;DR

This paper investigates finite size effects in random matrix theory related to Riemann zeros, analyzing data thinning effects and deriving differential equations with Painlevé transcendents to model spacing corrections, showing strong agreement with empirical data.

Contribution

It introduces a method to analyze thinning of Riemann zero data and derives a differential equation with Painlevé transcendents to characterize finite size corrections.

Findings

Thinning the data affects two-point correlation functions.

Derived differential equation models spacing corrections.

Numerical solutions agree with Riemann zero data.

Abstract

Odlyzko has computed a data set listing more than $10^9$ successive Riemann zeros, starting at a zero number beyond $10^{23}$. The data set relates to random matrix theory since, according to the Montgomery-Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and collaborators, have used $N \times N$ random unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the data set is to thin it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterise the order $1/N^2$ correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are Painlev\'e transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. Comparison with the Riemann zero data shows accurate agreement.