Numerical study of the derivative of the Riemann zeta function at zeros (Q2910114)

Let \(\zeta(s)\) denote the Riemann zeta-function, and let \(\gamma_n>0\) denote the ordinate of \(n\)th nontrivial zero of \(\zeta(s)\). Assuming the Riemann hypothesis (RH), the author investigates numerically the distribution of the derivative of the Riemann zeta-function at the zeros of \(\zeta(s)\).NEWLINENEWLINEAssuming RH and weak consequence of Montgomery's pair correlation conjecture, \textit{D. A. Hejhal} [in: Number theory, trace formulas and discrete groups, Symp. in Honor of Atle Selberg, Oslo/Norway 1987, 343--370 (1989; Zbl 0665.10027)] proved that, appropriately normalized, \(\log|\zeta'(\frac{1}{2}+i\gamma)|\) is normally distributed. In order to study the tails of the distribution of \(\log|\zeta'(\frac{1}{2}+i\gamma)|\) the authors study the moments NEWLINE\[NEWLINE J_\lambda(T) = \frac{1}{N(T)} \sum_{0<\gamma_n \leq T} \big| \zeta^\prime(1/2+i\gamma_n) \big|^{2\lambda} NEWLINE\]NEWLINE where \(N(T) =\#\{ 0<\gamma_n\leq T \}\) is the usual zero counting function. These moments have been extensively studied, both rigorously and conjecturally, initially by \textit{S. M. Gonek} [Invent. Math. 75, 123--141 (1984; Zbl 0531.10040); Mathematika 36, No. 1, 71--88 (1989; Zbl 0673.10032)] and later by a number of other authors. Modeling the zeta-function and its derivative using the characteristic polynomials from the classical compact groups, Hughes, Keating and O'Connell [\textit{C. P. Hughes} et al., Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 456, No. 2003, 2611--2627 (2000; Zbl 0996.11052)] have conjectured that NEWLINE\[NEWLINE J_\lambda(T) \sim a(k) \frac{G^2(\lambda+2)}{G(\lambda+3)} \left(\log\frac{T}{2\pi}\right)^{\lambda(\lambda+2)} NEWLINE\]NEWLINE for \(\mathrm{Re}(\lambda)>-3/2\) as \(T\to\infty\) where \(G(\cdot)\) is the Barnes \(G\)-function and \(a(k)\) is a certain ``arithmetic factor''. More recently, using the ratios conjecture, \textit{J. B. Conrey} and \textit{N. C. Snaith} [Proc. Lond. Math. Soc. (3) 94, No. 3, 594--646 (2007; Zbl 1183.11050)] made the more precise conjecture that NEWLINE\[NEWLINE J_\lambda(T) \sim \int_0^T P_\lambda \left(\log\frac{t}{2\pi}\right) \, \mathrm{d}t NEWLINE\]NEWLINE for certain polynomials \(P_\lambda(\cdot)\) when \(\lambda=1\) or 2, though it is likely their heuristics extend to all \(\lambda\in \mathbb{N}\).NEWLINENEWLINEThe authors numerically test these conjectures and the rate of convergence to various rigorously proved results, carefully explaining their methods and their results.NEWLINENEWLINEConclusions: ``Numerical data from high zeros of the zeta function generally agrees well with the asymptotic results that have been proved, as well as with several conjectures. There are some systematic differences between observed and expected distributions, but the discrepancies decline with growing heights.NEWLINENEWLINEThe results of this paper provide additional evidence for the speed of convergence of the zeta function to its asymptotic limits. They also demonstrate the importance of outliers, and thus the need to collect extensive data in order to obtain valid statistical results. The long-range correlations that have been found among values of the derivative of the zeta function at zeros can be explained by known analytic techniques.''