Estimate of the logarithmic derivative of the Riemann zeta function

Question

How I can achieve this result. If $\sigma > 1$ $$-\frac{\zeta'}{\zeta}(\sigma) \ll (\sigma -1 )^{-1}$$

Thanks!

Answer 1

Zeta'[x] / Zeta[x] has negative values for all x > 1. WHen x goes to infinity, its limit is zero by negative values. The right hand side is always positive.