Density results for $r$-gaps between zeros of the Riemann zeta-function

Abstract

Let $0<{\gamma }_1\le {\gamma }_2\le \dots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $\Theta ,\vartheta >0$ such that for each $r\in \mathbb{N}$, \[ \limsup_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\geq 1+\fracΘ{r^α} \qquad \text{and}\qquad \liminf_{n\to \infty}\frac{γ_{n+r}-γ_n}{2πr/\log γ_n}\leq 1-\frac{\vartheta}{r^α} \] where $\alpha$ may be taken as $2/3$, or as $1/2$ if one assumes the Riemann hypothesis. This was recently proved by Conrey and Turnage-Butterbaugh under RH and by Inoue unconditionally. We prove that in fact a positive proportion of $r$-gaps are large (and small) to the above extent, and we provide explicit estimates for the sizes and proportions of these gaps. In the case $r=1$, this quantitatively improves an unconditional result of Simonič, Trudgian and Turnage-Butterbaugh.