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Some remarks on the mean-value of the Riemann zeta-function and other Dirichlet series. IV. (English) Zbl 0882.11049

[For parts I, II, and III see Hardy-Ramanujan J. 1, 1-15 (1978; Zbl 0411.10013); ibid. 3, 1-24 (1980; Zbl 0426.10046); and Ann. Acad. Sci. Fenn., Ser. A I 5, 145-158 (1980; Zbl 0448.10031).]
This paper is concerned with \[ \max_{\sigma\geq\alpha} \Biggl({1\over H} \int^{T+ H}_T \Biggl|{d^m\over ds^m} (\zeta(s))^{2k}\Biggr|dt\Biggr) \] when \(k>0\) and \({1\over 2}<\alpha\leq 2\). When \(T\geq H\gg_{k,m}1\) and also (if \(2k\not\in\mathbb{N}\)) \(0\leq m\leq 2k\), a lower bound \(\gg_{k,m}(\alpha-{1\over 2})^{-k^2- m}\) is obtained for \(\alpha\geq{1\over 2}+{q\over\log H}\). Here \(q\) is the denominator of a suitable rational approximation to \(k\). The lower bound obtained is of the expected order of magnitude. It would be nice to prove the corresponding result for \(\sigma=\alpha\).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M41 Other Dirichlet series and zeta functions
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