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

Abstract

Let $0<\gamma_1\leq \gamma_2\leq \ldots$ 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 $r\in \mathbb{N}$, \[ \limsup_{n\to \infty}\frac{\gamma_{n+r}-\gamma_n}{2\pi r/\log \gamma_n}\geq 1+\frac{\Theta}{r^\alpha} \qquad \text{and}\qquad \liminf_{n\to \infty}\frac{\gamma_{n+r}-\gamma_n}{2\pi r/\log \gamma_n}\leq 1-\frac{\vartheta}{r^\alpha} \] 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 Simoni\v{c}, Trudgian and Turnage-Butterbaugh.