into Eqs. (99) and (100) and thereby obtain other expressions. This too we omit.

**An extension and brief discussion**

In principle, a result of the shape of Proposition 1 holds for all convergent Dirichlet series. Rather than describe this generally, we illustrate this for a Dirichlet series of particular interest. Namely, for $${{\zeta'} \over {\zeta}}(s)=-\sum_{k=2}^\infty {{\Lambda(k)} \over k^s},
~~~~~~~~~~\mbox{Re} ~s >1, \eqno(116)$$ where $\Lambda$ is the von Mangoldt function [@ivic; @titch]. Like $\zeta$ itself, the logarithmic derivative (116) has a simple pole at $s=1$ and the Laurent expansion analogous to Eq. (1) is $${{\zeta'(s)} \over {\zeta(s)}}=-{1 \over {s-1}}-\sum_{p=0}^\infty 
\eta_p (s-1)^p, ~~~~~~|s-1| < 3, \eqno(117)$$ wherein the constants $\eta_j$ can be written as $$\eta_k={(-1)^k \over {k!}} \lim_{N \to \infty}\left (\sum_{m=1}^N {1 \over 
m} \Lambda(m)\ln^k m - {{\ln^{k+1} N} \over {k+1}}\right ). \eqno(118)$$ The $\eta_j$ coefficients are related to the Stieltjes constants $\gamma_k$ by a simple recurrence relation (e.g., [@coffeyjcam04], Appendix).

We now have . For integers $j \geq 0$ we have $$\eta_j = -{1 \over {j!}}\sum_{m=2}^\infty {{(-1)^m} \over m}\left ({{\zeta'}
\over \zeta}\right)^{(j)}(m)+{{(-1)^j} \over {j!}}\lim_{N \to \infty}
\left[\sum_{k=2}^N \Lambda(k) \ln^j k \ln\left({{k+1} \over k}\right)-
{{\ln^{j+1} N} \over {j+1}}\right], \eqno(119)$$ and . We have $$\eta_0=-\gamma = -\sum_{m=2}^\infty {{(-1)^m} \over m}{{\zeta'}
\over \zeta}(m)+\lim_{N \to 