Problem: Formulate a precise and believable statement of the ``GUE Hypothesis.''
Any reasonable form of the conjecture should predict that the correlation functions of the zeros and the distribution of the neighbor spacings of the zeros of any automorphic L-function should have the same statistics as the eigenvalues of some collection of matrices.
The question is: which statistics? what collection of matrices (which may be a function of how `high up' the zeros are)? what is the rate of convergence, and how uniform is it in the various parameters?
Montgomery's original pair correlation conjecture was that if then as , uniformly for . A weak version of the conjecture asserts that , uniformly for . Such an ``almost everywhere'' version is usually all that is needed for applications. Goldston and Montgomery [ MR 90h:11084] show that it is equivalent to a statement about the variation of the distribution of primes in short intervals, and Goldston, Gonek, and Montgomery (to appear in Crelle) show tht it is equivalent to a statement about the mean square of near the -line.
It is possible that an acceptable formulation of the GUE conjecture can be made in terms of integrals of ratios of $\zeta$-functions.
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