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Recall that the Stieltjes constants $\gamma_{k}$ appear as the coefficients in the regular part of the Laurent expansion of the Riemann zeta function about $s = 1$:
$$ \begin{align} \zeta(s) = \frac{1}{s-1}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!} \gamma_{k}\:(s-1)^{k}, \quad \,s\neq 1, \tag1 \end{align} $$ with $\displaystyle \gamma_{0}=\gamma$, the Euler constant.

Similarly the generalized Stieltjes constants $\gamma_{k}(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function about $s = 1$:
$$ \begin{align} \zeta(s,a) = \frac{1}{s-1}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!} \gamma_{k}(a)\:(s-1)^{k}, \quad a>0, \,s\neq 1, \tag2 \end{align} $$ with $\displaystyle \gamma_{0}(a)=-\psi(a)$, the digamma function.

I am aware of a theorem of W. E. Briggs (1955) saying that infinitely many Stieltjes constants $\gamma_{k}$ take positive values, and infinitely many take negative values and I am aware of Mitrovic's theorem in the same vein. I've read Mark Coffey's papers.

I'm looking for any interesting material/references about the sign of $\gamma_{k}(a),\,a>0$.

Your help is welcome.

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