# What is the effective lower bound on gaps between zeta zeros?

In this question here: Upper bound on differences of consecutive zeta zeros by Charles it is said that: "There are many papers giving lower bounds to:

$$\limsup_n\ \delta_n\frac{\log\gamma_n}{2\pi}$$

unconditionally or on RH or GRH." RH stands of course for the Riemann hypothesis.

Therefore I am asking: What is the best unconditional effective lower bound for gaps $$\delta_n=|\gamma_{n+1}-\gamma_n|$$ between consecutive non-trivial Riemann zeta function zeros?

## 1 Answer

In 2012 Feng and Wu showed that $$\limsup_{n} \delta_n \frac{\log \gamma_n}{2\pi}\geq 2.7327.$$

Note however that the quantity $$\mu=\liminf_n\delta_n \frac{\log \gamma_n}{2\pi}$$ is far more interesting. Unconditionally, the best bound is $\mu<0.525396$ due to this recent paper of Preobrazhenskii, however even under the Riemann Hypothesis this cannot be improved very much. Proving that $\mu<1/2$ would be a remarkable achievement, even under the Riemann hypothesis, as this would resolve the longstanding conjecture that there are no Siegel Zeros for any Dirichlet L-function. This remarkable connection between the spacing of the zeros of $\zeta(s)$, and the properties of the zeros of all Dirichlet L-functions is known as the Deuring–Heilbronn phenomenon.

• Since you know of the paper Feng and Wu, that I was aware of too, but I have not read in its entirety, What is meant with the phrase in the abstract: "...,we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least..."? Does "infinitely often" equal always? Repeat of my question here: Is "infinitely often" the same as saying: "...consecutive non-trivial zeros of the Riemann zeta-function differ always by at least..." – Mats Granvik Nov 25 '15 at 14:10
• Also. I am not yet competent in understanding the notions "sup" and "inf". – Mats Granvik Nov 25 '15 at 14:21