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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 $$1 \leq k \leq {\log\log T\over 2\log\log\log T}$$.

##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$
##### Keywords:
Riemann $$\xi$$-function; zeros of derivatives
Zbl 0502.10022
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