On r ‐gaps between zeros of the Riemann zeta‐function

Under the Riemann Hypothesis, we prove for any natural number r there exist infinitely many natural numbers n such that (γn+r−γn)/(2πr/logγn)>1+Θ/r and (γn+r−γn)/(2πr/logγn)<1−ϑ/r for explicit absolute positive constants Θ and ϑ , where γ denotes an ordinate of a zero of the Riemann zeta‐function on the critical line. Selberg published announcements of this result several times without proof.