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On negative moments of the Riemann zeta-function. (English) Zbl 0673.10032

This paper is concerned with the negative moments \[ I_{- k}(\sigma,T)=\int^{\infty}_{0}| \zeta (\sigma +it)|^{-2k} dt\quad and\quad J_{-k}(T)=\sum_{0<\gamma \leq T}| \zeta '(\rho)|^{-2k} \] where \(k>0\) and \(\sigma\geq 1/2\). Assuming the Riemann Hypothesis it is shown that \[ I_{-k}(\sigma,T)\quad \gg_ k\quad T(\min (\frac{1}{\sigma -}, \log T))^{k^ 2} \] uniformly for \(\sigma\leq 1-\delta\), and that if, in addition, all zeros of \(\zeta\) (s) are simple, then \(J_{-1}(T)\gg T.\)
A number of conjectures, supported by heuristic arguments, are also given. The proofs of the main results use a “Bessel’s inequality” argument, modelled on the work of J. B. Conrey and A. Ghosh [Mathematika 31, 159-161 (1984; Zbl 0528.10026)].
Reviewer: D.R.Heath-Brown

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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References:

[1] Titchmarsh, The Theory of the Riemann Zeta-function (1986) · Zbl 0601.10026
[2] Hejhal, Number Theory, Trace Formulas and Discrete Groups, Symposium in Honor ofAtle Selberg, Oslo, Norway, July pp 343– (1989) · doi:10.1016/B978-0-12-067570-8.50027-5
[3] Conrey, Mathematika 31 pp 159– (1984)
[4] DOI: 10.1007/BF01403094 · Zbl 0531.10040 · doi:10.1007/BF01403094
[5] DOI: 10.1112/jlms/s2-24.1.65 · Zbl 0431.10024 · doi:10.1112/jlms/s2-24.1.65
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