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On the frequency of Titchmarsh’s phenomenon for \(\zeta\) (s). VI. (English) Zbl 0649.10028
Let \(E>1\) be a fixed constant, \(C\leq H\leq T/100\) and \(K=Exp((D \log H)/(\log \log H))\) where C is a large positive constant and D an arbitrary positive constant. Then the main result is as follows. Theorem: There are \(\geq TK^{-E}\) disjoint intervals I of length K each and all contained in [T,2T] such that the maximum of \(| \zeta (1+it)|\) as t varies over I lies between \[ e^{\gamma}(\log \log K-\log \log \log K+O(1))\quad and\quad e^{\gamma}(\log \log K+\log \log \log K+O(1)). \] In the proof of this theorem one of the tools is the main theorem of part V of this series [Ark. Mat. 26, No.1, 13-20 (1988)]. The authors also announce a forthcoming result by the reviewer regarding the maximum of \(| \zeta (1+it)|\) over intervals I (contained in [T,2T]) of lengths \(\geq C \log \log \log \log T\) and smaller intervals. Here a precise lower bound is given for intervals of length \(\geq C \log \log \log \log T\) and statistical results for intervals of smaller lengths.
Reviewer: K.Ramachandra
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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