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. ) bywhere K is the integral operator on the interval with kernel Using an eigenvalue expansion of the Fredholm determinant, can be computed  (see also ) 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 , 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  for .