Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann’s zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many papers giving lower bounds to
$$
\limsup_n\ \delta_n\frac{\log\gamma_n}{2\pi}
$$
unconditionally or on RH or GRH. (The true value is believed to be $+\infty.$) I’m interested in an upper bound on the smaller quantity $\delta_n$. I asked the question on MathOverflow but have not yet found an effective bound. Both unconditional results and those relying on the RH are interesting.

Solutions Collecting From Web of "Upper bound on differences of consecutive zeta zeros"

There are two things:

1) On RH you have the clean bound
$$
\delta_n \leq \pi ( 1 + o(1)) / \log\log \gamma_n
$$
as $n \rightarrow \infty$,
due to Goldston and Gonek. If the $o(1)$ bothers you, you can remove it
by re-working the details in their (short) paper (see http://www.math.sjsu.edu/~goldston/article38.pdf in particular see Corollary 1).

Unconditionally, you have the point-wise bound due to Littlewood,
$$
\delta_n \leq C / \log\log\log \gamma_n
$$
I am not aware of anybody working out the explicit value of the constant $C$ in this case.

2) You can get better bounds if you are interested in bounds valid for “most” zeros. For example it is known that
$$
\sum_{T \leq n \leq 2T} \delta_n^{2k} \asymp T (\log T)^{-2k}
$$
This allows you to get good bounds for most $\delta_n$’s which are as good as
$\Psi(\gamma_n) / \log \gamma_n$ with a $\Psi(x)$ going to infinity arbitrarily slowly.