Research proof index

The logarithmic derivative of the Riemann zeta function is bounded in the half-plane Re(s) >= 3/2.

Let $0<r<R$, and let $B$ be analytic and nonvanishing on the closed disc $|z|\le R$. Then there is an analytic function $J_B$ on $|z|<R$ with $J_B(0)=0$, $J_B'(z)=\frac{B'(z)}{B(z)}\quad(|z|\le r),$ and $\operatorname{Re}J_B(z)=\log|B(z)|-\log|B(0)|\quad(|z|<R).$ Thus $J_B$ is a normalized analytic logarithm of $B/B(0)$.