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On the distribution of \(\log | \zeta '(+it)|\). (English) Zbl 0665.10027

Number theory, trace formulas and discrete groups, Symp. in Honor of Atle Selberg, Oslo/Norway 1987, 343-370 (1989).
[For the entire collection see Zbl 0661.00005.]
It was shown by Selberg in unpublished work that log \(| \zeta (+it)|\) has a normal distribution, in the sense that \[ mes\{t\in [0,T]:\frac{\log | \zeta (+it)|}{\sqrt{\pi \log \log T}}\in I\}\to \int_{I}e^{-\pi x^ 2} dx \] as \(T\to \infty\), for any interval I. The present paper gives the corresponding result for \(\zeta '(+it)\). Specifically, let \(A(t)=(t/2\pi)\log (t/2\pi e)\), then, on the Riemann Hypothesis one has \[ mes\{t\in [0,T]:\frac{\log | \zeta '(+it)/A'(t)|}{\sqrt{\pi \log \log T}}\in I\}\to \int_{I}e^{-\pi x^ 2} dx \] as before. A similar result holds with \(\zeta '(+it)\) replaced by \(Z'(t)\) (in the usual notation of zeta-function theory). The proof is based on the heuristic expectation that \[ \log | \frac{\zeta '(+it)}{\zeta (+it)A'(t)}| \quad (=D(t),\quad say) \] should be O(1) for most t, and indeed the principal lemma is that \(\int^{T}_{0}| D(t)|^{2k} dt=O_ k(T)\) for any positive integer k. The argument is distinctly more complicated than that used for Selberg’s result. It builds on the same ideas, but requires more careful consideration of the spacing between zeros.
Reviewer: D.R.Heath-Brown

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms

Citations:

Zbl 0661.00005