Neighbor spacing of zeros of L-functions

We use the notation from Distribution of zeros of L-functions, and everything below assumes RH and is specialized to the case of the Riemann $\zeta$-function.

Write $d_j=\tilde\gamma_{j+1}-\tilde\gamma_j$ for the normalized difference between consecutive zeros of the $\zeta$-function. The GUE conjectures imply that for all $a,A>0$ we have $d_j<a$ for a positive proportion of $j$, and $d_j>A$ for a positive proportion of $j$. There have been a number of efforts aimed at showing $a<\frac12$ because this would prove the nonexistence of Siegel zeros. (See [49 #2590] for a reference). At present the best results, which are due to Soundararajan [ MR 97i:11097], are $a<0.6878$ and $A>1.4843$.

The GUE conjectures also imply that for all $\mu,\lambda>0$ we have $d_j<\mu$ for infinitely many $j$, and $d_j>\lambda$ for infinitely many $j$. At present the best results (which assume RH and GLH) are [ MR 88g:11057] $\lambda>2.68$ and [ MR 86i:11048] $\mu<0.5172$. Unconditionally, Richard Hall (unpublished) has shown $\lambda > \sqrt{\frac{11}{2}}=2.345207\ldots$.

It has not been shown that $\mu<\frac12$ implies the nonexistence of Siegel zeros. However, Conrey and Iwaniec have recently shown that $d_j < \frac12 - \delta$, for $\gg T/\log^A T$ zeros with $0<\gamma<T$, implies the nonexistence of Siegel zeros.

As described in the article on The Alternative Hypothesis, the possibility of $d_j\ge \frac12$, for all $j$, is consistent with everything which is known about the correlation functions of the zeros of the $\zeta$-function. However, it is possible that there is a $C>\frac 12$ such that $d_j\ge C$, for all $j$, is also consistent with current information on the correlation functions. It would be interesting to know the correct answer.

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