Zeros of the Riemann zeta-function on the critical line (Q1319284)

It is known that \(\zeta(s)\) has infinitely many zeros of odd order on the critical line. Write these in the form \(1/2+ i\widehat{\gamma}_ n\) where \(0<\widehat {\gamma}_ 1<\widehat {\gamma}_ 2<\ldots\). It was shown by \textit{A. Selberg} [Skr. Nor. Vidensk. Akad., Oslo 10, 1-59 (1942; Zbl 0028.11101)] that the number of such points up to \(T\) is of exact order \(T\log T\). By a refinement of Selberg's argument it is now proved that \[ \sum_{\widehat {\gamma}_ n\leq T} (\widehat {\gamma}_{n+1}-\widehat {\gamma}_ n)\ll_ \mu T(\log T)^{1-\mu} \] for any constant \(\mu\in(0,2)\). It follows that if \(f(T)\) is any function which tends to infinity with \(T\) then \(\zeta(1/2+it)\) has a zero of odd order in ``almost all'' intervals \([T,T+f(T)(\log T)^{-1}]\).