A note on $r$-gaps between zeros of the Riemann zeta-function

Abstract

In this note, we prove Selberg's announced result on $r$-gaps between zeros of the Riemann zeta-function $\zeta$. Our proof uses a result on variations of $\arg\zeta$ by Tsang based on Selberg's method. The same result with explicit constants under the Riemann Hypothesis has been obtained by Conrey and Turnage-Butterbaugh using a different method. We explain how to obtain explicit constants under the Riemann Hypothesis using our approach which is based on Selberg's and Tsang's arguments.