On the frequency of Titchmarsh's phenomenon for \(\zeta\) (s). V (Q1113933)

The following \(\Omega\)-estimate of statistical nature is proved. Let \(\alpha\) be a constant with \(<\alpha <1\), \(E>1\) an arbitrary constant, C a large positive constant, D any positive constant, \(C\leq H\leq T/100\), and \(K=\exp (D \log H/\log \log H)\). Then there are at least \(TK^{-E}\) disjoint intervals I of length K each contained in [T,2T] such that the maximum of \(| \log (\zeta (\alpha +it))|\) over I lies between constant multiples of \((\log K)^{1-\alpha}/(\log \log K)^{\alpha}.\) This paper is related to the one with number II of the present series, written by \textit{K. Ramachandra} [Acta Math. Acad. Sci. Hung. 30, 7-13 (1977; Zbl 0373.10025)].