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 -function.
Write for the normalized difference between consecutive zeros of the -function. The GUE conjectures imply that for all we have for a positive proportion of , and for a positive proportion of . There have been a number of efforts aimed at showing 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 and .
The GUE conjectures also imply that for all we have for infinitely many , and for infinitely many . At present the best results (which assume RH and GLH) are [ MR 88g:11057] and [ MR 86i:11048] . Unconditionally, Richard Hall (unpublished) has shown .
It has not been shown that implies the nonexistence of Siegel zeros. However, Conrey and Iwaniec have recently shown that , for zeros with , implies the nonexistence of Siegel zeros.
As described in the article on The Alternative Hypothesis, the possibility of , for all , is consistent with everything which is known about the correlation functions of the zeros of the -function. However, it is possible that there is a such that , for all , is also consistent with current information on the correlation functions. It would be interesting to know the correct answer.
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