The signs of the Stieltjes constants associated with the Dedekind zeta function

The Stieltjes constants ${\gamma }_n(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function ${\zeta }_K(s)$ at its pole $s=1$. In this paper, we establish a similar expression of ${\gamma }_n(K)$ as Stieltjes obtained in 1885 for $\gamma_n(\mathbb Q)$\mathrm{.}Wealsostudythesignsof$\gamma_n(K)$.