Question

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.

Answer 1

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

Answer 2

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 $\varepsilon$.