There are many results on the spacing of the gaps between nontrivial zeros of the $\zeta$ function, from trivial (average value is $\frac{2\pi}{\log\gamma\_n}$) to difficult (bounds on max and min values of the normalized gap). Are any reasonable upper bounds known? I'd like to have something that says, given any $\varepsilon>0,$ there is some N beyond which the gaps $\gamma\_{n+1}-\gamma\_n$ is at most $\varepsilon.$ This seems a weak request given the asymptotic behavior but I haven't found anything along these lines.


Any ideas?


I [asked the question on math.se](https://math.stackexchange.com/q/95693/1778) but did not get an answer.



Upper bounds on the difference of consecutive zeta zeros
Littlewood was the first to prove that the gaps between the ordinates of successive zeros of $\zeta(s)$ tend to zero. This is proved, for instance, in Titchmarsh's book on the zeta-function (see Theorem 9.11).


I believe the best known unconditional result states that
$$ \gamma\_{n+1}-\gamma\_n = O( 1/\log\log\log \gamma\_n)$$
as $n\to \infty$. Assuming the Riemann Hypothesis, this can be improved to $O( 1/\log\log \gamma\_n).$


