Random matrix theory and the derivative of the Riemann zeta function (Q2748061)

Define NEWLINE\[NEWLINEJ_k(T)= \frac{1}{N(T)} \sum_{0< \gamma_n\leq T} \bigl|\zeta' (\tfrac 12+ i\gamma_n) \bigr|^{2k},NEWLINE\]NEWLINE where \(\gamma_n\) is the coordinate of the \(n\)th nontrivial zero of \(\zeta(s)\). This has been previously studied by \textit{S. M. Gonek} [Mathematika 36, 71--88 (1989; Zbl 0673.10032)], amongst others. It has been proposed that the zeros \(\frac 12+ i\gamma_n\) may be modeled, under the Riemann Hypothesis, by the arguments of the eigenvalues \(\exp(i\theta_n)\) of random \(N\times N\) unitary matrices, with \(N\) around \(\log (\gamma_n/2\pi)\) [see \textit{J. P. Keating} and \textit{N. C. Snaith}, Commun. Math. Phys. 214, 57--89 (2000; Zbl 1051.11048)]. Specifically the present paper conjectures that NEWLINE\[NEWLINEJ_k(T)\sim a(k) \int_{U(N)} \frac 1N \sum_{n=1}^N|Z'(\theta_n)|^{2k} d\mu_N,NEWLINE\]NEWLINE where \(N=\log T/2\pi\), NEWLINE\[NEWLINEa(k)= \prod_p \Biggl(1- \frac 1p\Biggr)^{k^2} \sum_{m=0}^\infty \Biggl( \frac{\Gamma(m+k)} {m!\Gamma(k)} \Biggr)^2 p^{-m},NEWLINE\]NEWLINE and \(Z(\theta)= \prod_1^N (1-\exp (i\theta_n-\theta))\). Here the integration is over the unitary group \(U(N)\), with Haar measure \(d\mu_N\). The main substance of the paper however is the proof that NEWLINE\[NEWLINE\int_{U(N)} \frac{1}{N} \sum_{n=1}^N |Z'(\theta_n) |^{2k} d\mu_N\sim \frac{G^2(k+2)} {G(2k+3)} N^{k(k+2)},NEWLINE\]NEWLINE where \(G(z)\) is the Barnes \(G\)-function. NEWLINENEWLINENEWLINEThe paper also gives a detailed discussion of the distribution of \(\log|Z'(\theta_1)|\), where the eigenvalues \(\exp(i\theta_n)\) are ordered with \(0\leq \theta_1\leq \theta_2\leq\dots\leq 2\pi\). This establishes a direct analogue of \textit{D. A. Hejhal}'s result [Number theory, trace formulas, and discrete groups, Proc. 1987 Selberg Sympos., 343--370 (1989; Zbl 0665.10027)] on the distribution of \(|\zeta' (1/2+ i\gamma_n)|\).