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On simple zeros of derivatives of the Riemann \(\xi\)-function. (English. Russian original) Zbl 1131.11056
Izv. Math. 70, No. 2, 265-276 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 2, 57-68 (2006).
This paper is a continuation of the author’s earlier work [Izv. Math. 69, No. 3, 539–605 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 3, 109–178 (2005; Zbl 1102.11047)]. Let \(\xi(s)={s(s-1)\over 2}\pi^{-{s\over 2}}\Gamma({s\over 2})\zeta(s)\). It is easily seen from Weierstrass factorization that for every integer \(k\geq 0\), the zeros of \(\xi^{(k)}(s)\) lie in the strip \(0 < \operatorname{Re}\,s < 1\), and the Riemann Hypothesis implies that all of these zeros are on the line \(\operatorname{Re}\,s = {1\over 2}\). Let \(N_{k}(T)\) be the number of zeros of \(\xi^{(k)}(s)\) with \(0 < \operatorname{Im}\,s \leq T\), and let \(N_{k}'(T)\) be the number of simple zeros of \(\xi^{(k)}(s)\) with \(0 < \operatorname{Im}\,s \leq T\) and \(\operatorname{Re}\,s = {1\over 2}\). For \(T,U > 0\) define
\[ a_{k}'(T,U) = {N_{k}'(T+U)-N_{k}'(T)\over N_{k}(T+U)-N_{k}(T)}. \]
The result of this paper is that for every sufficiently large \(T\), taking \(U=T(\ln {T\over 2\pi})^{-10}\), one has
\[ a_{k}'(T,U) \geq 1 -{e^2 +2\over 16 k^2} \]
for every integer \(k \in [1,{\ln\ln T\over 2\ln\ln\ln T}]\). Formerly J. B. Conrey’s result [J. Number Theory 16, 49–74 (1983; Zbl 0502.10022)]
\[ \liminf_{T\to\infty} a_{k}'(T,U) \geq 1-O\biggl({1\over k^2}\biggr) \qquad (k\to\infty), \] with the same value of \(U\) as above, was known. The proof employs methods from the works of Conrey and of the author cited above.

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