  Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta
zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov,
Rubinstein and Snaith, which states that for any positive integer $K$ and real
number $a>0$, \begin{align*} \lim_{a \to 0^+}\lim_{T \to \infty}
\frac{(2a)^{2K-1}}{T (\log T)^{2K}} \int_{T}^{2T}
\left|\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\frac{a}{\log
T}+it\right)\right|^{2K} dt = \binom{2K-2}{K-1}. \end{align*} When $K=1$, this
was essentially a result of Goldston, Gonek and Montgomery.
