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On the zeros of derivatives of the Riemann \(\xi\)-function. (English. Russian original) Zbl 1102.11047
Izv. Math. 69, No. 3, 539-605 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 3, 109-178 (2005).
The Riemann \(\xi\)-function, \(\xi(s)={s(s-1)\over 2}\pi^{-{s\over 2}}\Gamma({s\over 2})\zeta(s)\) is an entire function of order one, and its zeros are the nontrivial zeros of the Riemann zeta-function \(\zeta(s)\). Let \(N_{k}(T)\) denote the number of zeros of \(\xi^{(k)}(s)\) with \(0 < \operatorname{Im} \rho \leq T\), and \(N_{k}^{(0)}(T)\) denote the number of such zeros which also satisfy \(\operatorname{Re} \rho = {1\over 2}\). For \(T,\, U > 0\) define \[ a_{k}(T,U) = {N_{k}^{(0)}(T+U)- N_{k}^{(0)}(T)\over N_{k}(T+U)-N_{k}(T)}. \] The Riemann Hypothesis is equivalent to the equality \(a_{0}(T,U)=1\). J. B. Conrey [J. Number Theory 16, 49–74 (1983; Zbl 0502.10022)] showed that \[ \liminf_{T\to \infty}a_{k}(T,{T\over (\log {T\over 2\pi})^{10}}) = 1 + O(k^{-2}), \] as \(k\to\infty\). In this paper the author refines this result to \[ a_{k}(T,{T\over (\log {T\over 2\pi})^{10}}) \geq 1 - {e^2 +2\over 16}k^{-2}, \] for all sufficiently large \(T\) and for every integer \(k\) with \(1 \leq k \leq {\log\log T\over 2\log\log\log T}\).

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
Zbl 0502.10022
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