Extreme gaps between consecutive zeros of the zeta-function

The number of zeros of $\zeta(s)$ with imaginary parts smaller than $T$ is given by

\begin{displaymath}N(T)=\frac{T}{2\pi}\log \frac{T}{2\pi e} +O(1/T) +S(T)\end{displaymath}


\begin{displaymath}S(T)=\frac{1}{\pi}\arg \zeta(1/2+iT).\end{displaymath}

It is known that $S(T)\ll \log T$ and, conditional on the Riemann Hypothesis that

\begin{displaymath}S(T) \ll \frac{\log T}{\log \log T}.\end{displaymath}

Thus, the average gap between zeros of $\zeta(s)$ at height $T$ is $\frac{2\pi}{\log T}$ and $S(T)$ measures the local fluctuations in the zero spacings. (If not for $S(T)$ the zeros of $\zeta(s)$ would have a `picket fence' spacing.)

Clearly, a large gap between consecutive zeros of $\zeta(s)$ implies that $S(T)$ is correspondingly large. It seems not unreasonable to speculate that the largest gaps between consecutive zeros of $\zeta(s)$ will `match' with the largest values of values of $S(T)$:

\begin{displaymath}\lim_{T\to \infty} \frac{(\log T)\max_{\gamma<T} (\gamma^+-\gamma)}{2 \pi \max_{t<T} S(t) }=1 \end{displaymath}

Thus, if $2 \pi S(T)$ is occasionally as large as $c \log T/\log \log T$ see the article on $S(T)$ then we would expect the maximal gaps between zeros of $\zeta$ to be as large as $c/\log \log T$.

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