One may recall that the Stieltjes constants $\gamma_{k}$ appear as the scaled coefficients in the regular part of the Laurent series expansion of the Riemann zeta function about $s = 1$:
$$
\begin{align} \zeta(s)
= \frac{1}{s-1}+\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!} \gamma_{k}\:(s-1)^{k}, \quad \,s \neq1, \tag1
\end{align}
$$ where $\displaystyle \gamma_{0}=\gamma$, the Euler-Mascheroni constant.
Do you know some result/reference connecting the sign of the Stieltjes constants and some zero-free region of the Riemann zeta function?
The same question holds for the Hurwitz zeta function $\zeta(\cdot,a)$ and the generalized Stieltjes constants $\gamma_{k}(a)$.
