Logarithmic derivative

The logarithmic derivative of a nonzero differentiable function $f$ is $f^{\prime }/f$. For $\Re s>1$, the Euler product for the Riemann zeta function gives

The classical zero-free region and a truncated Perron contour givefor some constant $c>0$.

If, for some real $u$,then partial summation continues $-{\zeta }^{\prime }(s-iu)/\zeta (s-iu)$ meromorphically to $\Re s>1/2$ with no pole except at $1+iu$. The zeta function has no zero to the right of the critical line, and its functional equation then implies the Riemann hypothesis.