  Assuming the Riemann hypothesis, we obtain a formula for the mean value of
the $k$-derivative of $\zeta'/\zeta$, depending on the pair correlation of
zeros of the Riemann zeta-function. This formula allows us to obtain new
equivalences to Montgomery's pair correlation conjecture. This extends a result
of Goldston, Gonek, and Montgomery where the mean value of $\zeta'/\zeta$ was
considered.
