On $r$-gaps between zeros of the Riemann zeta-function

Abstract

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $({\gamma }_{n+r}-{\gamma }_n)/(2\pi /\log {\gamma }_n)>r+\Theta \sqrt{r}$ and $({\gamma }_{n+r}-{\gamma }_n)/(2\pi /\log {\gamma }_n)<r-\vartheta \sqrt{r}$ for explicit absolute positive constants $\Theta$ and $\vartheta$, where $\gamma$ denotes an ordinate of a zero of the Riemann zeta-function on the critical line. Selberg published announcements of this result several times but did not include a proof. We also suggest a general framework which might lead to stronger statements concerning the vertical distribution of nontrivial zeros of the Riemann zeta-function.

to appear in the Bulletin of the London Mathematical Society

References in corpus (1)

• A note on the gaps between consecutive zeros of the Riemann zeta-function