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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
##### Keywords:
Riemann xi-function; zeros of derivatives
##### Citations:
Zbl 1102.11047; Zbl 0502.10022
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