Extreme gaps between consecutive zeros of the zeta-function
The number of zeros of with imaginary parts smaller than is given by
It is known that
and, conditional on the Riemann Hypothesis that
Thus, the average gap between zeros of at height is
and measures the local fluctuations in the zero spacings. (If not for the zeros of would have a `picket fence' spacing.)
Clearly, a large gap between consecutive zeros of implies that is correspondingly large. It seems not unreasonable to speculate that the largest gaps between consecutive zeros of will `match' with the largest values of values of :
Thus, if is occasionally as large as
see the article on $S(T)$ then we would expect the maximal gaps between zeros of to be as large as .
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