MATHPH
In the literature: open
Montgomery's GUE pair correlation of the Riemann zeros for all ranges
In plain words
Montgomery showed that the spacing between Riemann zeros matches random-matrix theory for some range of distances, assuming the Riemann hypothesis. Extending this to all distances is open.
Precise statement
Let $F(\alpha)$ be Montgomery's pair correlation function of normalized zeros. Prove $F(\alpha) = 1$ for $\mid\alpha\mid \ge 1$ (assuming RH), which gives the GUE pair correlation $1 - (\operatorname{sin}(\pi u)/(\pi u))^2$ for all test functions; it is proven only for Fourier support in $\mid\alpha\mid < 1$. Answer: a proof.
What would settle it
A proof of $F(\alpha) = 1$ for |alpha| >= 1, or of the pair correlation for test functions with Fourier support beyond $[-1, 1]$.