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Note on a paper by H. L. Montgomery (Omega theorems for the Riemann zeta- function). (English) Zbl 0743.11044

It is proved unconditionally that if \(\theta\) is fixed and \({1\over 2}<\sigma<1\) then \[ \hbox{Re}(e^{-i\theta}\log\zeta(\sigma+it))\geq C(\sigma)(\log t)^{1-\sigma}(\log\log t)^{-\sigma} \] for a sequence of arbitrarily large values of \(t\). Here \(C(\sigma)\gg 1/(1-\sigma)\) is an explicitly given constant, and one aim of the paper is to obtain as large a value as possible. On the Riemann Hypothesis a corresponding result for \(\sigma={1\over 2}\) is obtained. The authors follow the method used by H. L. Montgomery [Comment. Math. Helv. 52, 511-518 (1977; Zbl 0373.10024)].

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses

Citations:

Zbl 0373.10024
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