I don’t think there is a clean implication of this type. If one uses the explicit formula to try to work out whether there is a prime in the interval , one will find that one needs to understand a highly oscillatory sum that involves all the zeroes up to height about (up to log factors); even assuming the Riemann hypothesis, one would need to understand the cancellation in this sum. One could simply posit that this sum always exhibits the expected amount of cancellation, but this is a rather artificial conjecture which is basically just Cramer’s conjecture expressed in terms of the zeroes, and would be largely orthogonal to the GUE hypothesis which only concerns the local behaviour of zeroes rather than global cancellations.

]]>the whole page seems to repeat itself (twice!).

*[Fixed now – T.]*

I have a problem with this extract:

“… Thus the only remaining case is when {\chi} is real and {|t| 1 – \frac{c}{\log(q(2+|t|))} \}}. If there are two such zeroes …

The formula is incomplete …

*[Corrected, thanks – T.]*