A note on the mean values of the derivatives of $\zeta'/\zeta$

TL;DR

Under the Riemann hypothesis, the paper derives a formula for the mean value of derivatives of zeta'/zeta, linking it to zeros' pair correlation and providing new equivalences to Montgomery's conjecture.

Contribution

It extends previous work by deriving a formula for the mean values of derivatives of zeta'/zeta, connecting it to zeros' pair correlation and offering new equivalences to Montgomery's conjecture.

Findings

Derived a formula for the mean value of the k-derivative of zeta'/zeta.

Connected mean values to the pair correlation of zeros.

Provided new equivalences to Montgomery's pair correlation conjecture.

Abstract

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.