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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.

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