On the frequency of titchmarsh’s phenomenon for ζ(s)—III

We obtain a lower bound for max |ζ(1/2+it)| ast varies overT?t?T+Y, where (logT)^1/100?Y?T, as a function ofY(1/100 is unimportant). Our lower bound is exp {D(logY)^1/2 (log logY)^?1/2} whereD 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 caseY=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).