  We determine the strong coupling $\alpha_s(m_Z)$ from dimensionless ratios of
roots of moments of the charm- and bottom-quark vector and charm pseudo-scalar
correlators, dubbed
$R_q^{X,n}\equiv(M_q^{X,n})^\frac{1}{n}/(M_q^{X,n+1})^\frac{1}{n+1}$, with
$X=V,P$, as well as from the $0$-th moment of the charm pseudo-scalar
correlator, $M_c^{P,0}$. In the quantities we use, the mass dependence is very
weak, entering only logarithmically, starting at $\mathcal{O}(\alpha_s^2)$. We
carefully study all sources of uncertainties, paying special attention to
truncation errors, and making sure that order-by-order convergence is
maintained by our choice of renormalization scale variation. In the computation
of the experimental uncertainty for the moment ratios, the correlations among
individual moments are properly taken into account. Additionally, in the
perturbative contributions to experimental vector-current moments,
$\alpha_s(m_Z)$ is kept as a free parameter such that our extraction of the
strong coupling is unbiased and based only on experimental data. The most
precise extraction of $\alpha_s$ from vector correlators comes from the ratio
of the charm-quark moments $R_c^{V,2}$ and reads $\alpha_s(m_Z)=0.1168\pm
0.0019$, as we have recently discussed in a companion letter. From bottom
moments, using the ratio $R_b^{V,2}$, we find $\alpha_s(m_Z)=0.1186\pm0.0048$.
Our results from the lattice pseudo-scalar charm correlator agree with the
central values of previous determinations, but have larger uncertainties due to
our more conservative study of the perturbative error. Averaging the results
obtained from various lattice inputs for the $n=0$ moment we find
$\alpha_s(m_Z)=0.1177\pm0.0020$. Combining experimental and lattice information
on charm correlators into a single fit we obtain $\alpha_s(m_Z)=0.1170\pm
0.0014$, which is the main result of this article.
