On the frequency of Titchmarsh's phenomenon for ζ(s)-III

Abstract

We obtain a lower bound for max |ιζ(½+it)| as t varies overT < t < T+Y, where (log T)^1/100 < Y < T, as a function of Y(1/100 is unimportant). Our lower bound is exp {D(log Y)^½ (log log Y)^−½} where D is a positive constant. (After submitting this paper for publication we came to know through a preprint of H L Montgomery that he had proved our result in the case Y=T. In his proof an essential assumption is Riemann hypothesis and our result is independent of any such unproved hypothesis. However he has other new results which are free from any hypothesis).