Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$?
I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes
$$\frac{\zeta^\prime (s)}{\zeta(s)}= A+ (s-1)^{-1}$$
where $A$ is a real number constant.