TITLE: Logarithmic derivative of Riemann Zeta function
QUESTION [15 upvotes]: 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.

REPLY [6 votes]: As an addendum to Eric's answer, Coffey gives a recursion relation for the coefficients $\dfrac{\eta_j}{j!}$ appearing in the expansion of the $\zeta$ function's logarithmic derivative. Letting $\gamma_k$ be the Stieltjes constants, and with the initial condition $\eta_0=-\gamma_0=-\gamma$, we have the recursion relation
$$\frac{\eta_j}{j!}=(-1)^{j+1}\left(\frac{j+1}{j!}\gamma_j+\sum_{k=0}^{j-1} \frac{(-1)^{k-1}}{(j-k-1)!}\frac{\eta_k}{k!}\gamma_{j-k-1}\right)$$
A similar formula also appears in Choudhury's paper.