  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.
