Distribution of zeros of L-functions

For ease of notation we will phrase everything in terms of the Riemann $\zeta$-function, with the understanding that all statements hold for general $L$-functions with the obvious modifications.

Assume the Riemann Hypothesis, let $\frac12 + i\gamma$ denote a nontrivial zero of the $\zeta$-function with $\gamma>0$, and write $\gamma_j$ for the $j$th zero, ordered with increasing imaginary part and repeated according to their multiplicity. Let $\tilde\gamma := \frac{1}{2\pi}\gamma\log(\gamma)$, so that $d_j:=\tilde\gamma_{j+1}-\tilde\gamma_j $ has average value of 1. An important problem is establishing various statistical properties of the sequence of $\tilde\gamma$.

The pioneering work of Montgomery [49 #2590] on the pair correlation of zeros of the $\zeta$-function, work of Hejhal [ MR 96d:11093], and Rudnick and Sarnak [ MR 97f:11074], on higher correlations of zeros, and extensive numerical calculations of Odlyzko (see [ MR 88d:11082] and [unpublished work] available on his web page), give persuasive evidence of the following:

The GUE Conjecture The (suitably rescaled) zeros of the Riemann $\zeta$-function are distributed like the eigenvalues of large random matrices from the Gaussian Unitary Ensemble.

The conjecture has yet to be stated in a precise form. See The GUE hypothesis for a discussion. However, any reasonable form of the conjecture makes a prediction of the correlation functions of the zeros and the distribution of the neighbor spacings of the zeros, as well as various other statistics. Thus, the GUE conjecture is a powerful tool for making and testing conjectures about the $\zeta$-function, and for shedding new light on a variety of long-standing questions. See the discussions of mean values and value distribution for more examples of how this conjecture relates to classical objects studied in number theory.

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