# 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.

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.

• Come on give me points people, I worked on that answer and I answered his question! Oct 20 '12 at 4:47
• Let's get the points flowing Oct 20 '12 at 4:47
• Voting is a capricious thing.
– robjohn
Oct 20 '12 at 8:35
• Very nice! Absolutely deserving of the points. I am looking at extremal behavior of the gaps, so the first part is what interests me. I had not seen the paper. Oct 22 '12 at 13:18
• @ blabler and robjohn : How is it known : the equation from 2) with the sigma ? Voting capricious ? I disagree. The entire MSE is based upon voting ! And I feel sorry for blabler. He deserves more votes and I am also sorry for me. Why ? Because I lost a bounty to this question and he missed my bounty , so we both lost ! Now that bounty thing did not work and it has failed for me before , now that is capricious ! Are you in a bad mood perhaps ? Welcome to MSE blabler !
– mick
Oct 22 '12 at 16:15