The GUE hypothesis

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 $\alpha>1$ then $F(\alpha,T)\sim 1$ as $T\to \infty$, uniformly for $1\le \alpha \le A$. A weak version of the conjecture asserts that $\int_{\alpha}^{\beta} F(x,T)dx\sim \beta-\alpha$, uniformly for $1\le\alpha<\beta<A$. 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 $\zeta'/\zeta$ near the $\frac12$-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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