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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 but did not get an answer.

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2 Answers 2

up vote 9 down vote accepted

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 $O( 1/\log\log \gamma_n).$

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Do you know of an effective version of the first? –  Charles Jan 6 '12 at 13:26
    
This is Theorem 9.12 in Titchmarsh 'Theory of the Riemann Zeta Function.' The proof uses the Borel-Caratheodory theorem, and can be made effective if you really really want it. Titchmarsh has a series of seven successive constants $A_1, A_2, \ldots, A_6, A$, with the final $A$ being the constant in the big Oh term above. –  Stopple Jan 11 '12 at 22:07

From the standard zero-counting formula $N(T) = \frac{T}{2\pi} \log(\frac{T}{2 \pi e}) + O(\log{T})$, this shows $N(T + h) - N(T) = \frac{h}{2 \pi} \log(\frac{T}{2 \pi}) + O(\log{T})$, and hence $N(T+h) - N(T) \geq 1$ provided $h$ is large enough compared to the implied constants. This shows what you ask for with an unspecified $\varepsilon$.

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Yes, that was my first thought (though I must have do.e my arithmetic wrong since I found a larger error term). But can it be made effective? –  Charles Jan 5 '12 at 21:00
    
It can be made effective, and surely has been, but I don't know a source offhand. –  Matt Young Jan 5 '12 at 21:15
    
This also gives the desired result for some $\varepsilon$ even without the "some $N$ beyond which ...". Can it be proved for every $\varepsilon > 0$ (with $N\phantom.$ necessarily depending on $\varepsilon$)? –  Noam D. Elkies Jan 5 '12 at 21:54
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Matt, I am a bit confused. Aren't you only proving bounded gaps? –  Micah Milinovich Jan 6 '12 at 4:58
    
@Micah: yes, the question as stated asked for the existence of some $\varepsilon$ for which the gaps are bounded by $\varepsilon$. I know $\varepsilon$ usually wants to tend to zero though... –  Matt Young Jan 6 '12 at 18:20

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