Riemann zeta expansion about a=1

The Riemann zeta function is defined by the Dirichlet series

(1) $\begin{equation*}\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\end{equation*}$

as a sum of reciprocal natural numbers raised to power $s$ , which is absolutely convergent for $\Re(s)>1$ and defined for a complex variable $s=\sigma+it$ . Moreover, the $\zeta$ extends meromorphically to $\mathbb{C}$ having a simple pole at $s=1$ with residue $1$ , and there are many representations of $\zeta$ valid in different domains and many in all $\mathbb{C}\slash 1$ . The Laurent expansion about the pole is a common example

(2) $\begin{equation*}\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\gamma_n\frac{(-1)^n(s-1)^n}{n!}\end{equation*}$

of such a global representation, where its Stieltjes expansion coefficients are given by formula

(3) $\begin{equation*}\gamma_n=\lim_{k\to\infty}\Bigg{\sum_{m=1}^{k}\frac{\log^n(m)}{m}-\frac{\log^{n+1}(k)}{n+1}\Bigg}\end{equation*}$

as $k\to \infty$ , which is commonly written in the literature [3][3].