Finite-size corrections in random matrix theory and Odlyzko’s dataset for the Riemann zeros

Abstract

Odlyzko has computed a dataset listing more than 10 9 successive Riemann zeros, starting from a zero number to beyond 10 23 . This dataset relates to random matrix theory as, 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 co-workers, have used N × 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 dataset is to minimize 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 characterize 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 Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. In comparison to the Riemann zero data accurate agreement is exhibited.