Stieltjes constants

The Stieltjes constants are defined by $$\gamma_n = \displaystyle\lim_{m \rightarrow \infty} \left[ \displaystyle\sum_{k=1}^m \dfrac{\log^n(k)}{k} - \dfrac{\log^{n+1}(m)}{n+1} \right]$$

Properties

Theorem

The following Laurent series holds: $$\zeta(z)=\dfrac{1}{z-1} + \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k \lambda_k (z-1)^k}{k!},$$ where $\zeta$ denotes the Riemann zeta function and $\lambda_k$ denotes the Stieltjes constants.