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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
##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
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