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 $\gamma_n(\mathbb Q)$. We also study the signs of $\gamma_n(K)$.