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1. INTRODUCTION

The so called GUE hypothesis (see e.g. [1]) states that, in a certain limit, the zeros of the Riemann zeta function on the critical line Re

have the same joint distribution as that of the eigenvalues of a random matrix from the Gaussian Unitary Ensemble (GUE) of large (formally infinite) dimensional random Hermitian matrices. Denoting the zeros by

, where n labels the zeros sequentially along the critical line, the GUE hypothesis applies in the limit

, with each

scaled by the mean density of zeros at

, so that the mean spacing between zeros is unity (any finite value will do).

We recall (see e.g. [2]) that a random Hermitian

matrix is said to belong to the GUE if the diagonal elements

(which must be real) and the upper triangular elements

are independently chosen with probability density function (p.d.f.)

respectively. For large N the density of eigenvalues

is given by the so called Wigner semi-circle law

To apply the GUE hypothesis the eigenvalues should therefore be scaled by

before the

is taken, to obtain a mean eigenvalue spacing of unity.

One consequence of the GUE hypothesis is that it provides concrete predictions for statistical properties of , whenever these are known for the GUE random matrices. One such example is the p.d.f., say, for the spacing between consecutive zeros. In the infinite GUE, scaled so that the mean eigenvalue spacing is , the corresponding quantity is given in terms of a Fredholm determinant of an integral operator (see e.g. [2]) by

where K is the integral operator on the interval

with kernel

Using an eigenvalue expansion of the Fredholm determinant,

can be computed [3] (see also [1]) to give a tabulation or graph of

In a large-scale numerical computation of the non-trivial zeros of the Riemann zeta function by one of the present authors [1], involving over consecutive values of about calculated to an accuracy of about six decimal places, the p.d.f. has been determined empirically and compared with as calculated from (1.3). Excellent agreement is found. Similar agreement is found when comparing other empirical statistical distributions with those known exactly for the GUE.

In this paper we present the exact calculation of a new statistical quantity for the infinite GUE, which is similar in meaning to . This statistical quantity is the p.d.f., say, for the spacing between nearest neighbor eigenvalues. We then test the GUE hypothesis by comparing for the infinite GUE with the empirical calculation of for the zeros the Riemann zeta function on the critical line from the data of [1] for .

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