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Under the Riemann Hypothesis, we prove for any natural number   r   there exist infinitely many natural numbers   n   such that(   γ   n  +  r    −   γ  n   )   /   (  2  π  r  /  log   γ  n   )   >  1  +  Θ  /   rand(   γ   n  +  r    −   γ  n   )   /   (  2  π  r  /  log   γ  n   )   <  1  −  ϑ  /   rfor 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.

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