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We investigate a large class of perturbative QCD (pQCD) renormalizationschemes whose beta functions $\beta(a)$ are meromorphic functions of therunning coupling and give finite positive value of the coupling $a(Q^2)$ in theinfrared regime ("freezing"), $a(Q^2) \to a_0$ for $Q^2 \to 0$. Such couplingsautomatically have no singularities on the positive axis of the squared momenta$Q^2$ ($ \equiv -q^2$). Explicit integration of the renormalization groupequation (RGE) leads to the implicit (inverted) solution for the coupling, ofthe form $\ln (Q^2/Q^2_{\rm in}) = {\cal H}(a)$. An analysis of this solutionleads us to an algebraic algorithm for the search of the Landau singularitiesof $a(Q^2)$ on the first Riemann sheet of the complex $Q^2$-plane, i.e., polesand branching points (with cuts) outside the negative semiaxis. We presentspecific representative examples of the use of such algorithm, and compare thefound Landau singularities with those seen after the 2-dimensional numericalintegration of the RGE in the entire first Riemann sheet, where the latterapproach is numerically demanding and may not always be precise. The specificexamples suggest that the presented algebraic approach is useful to find outwhether the running pQCD coupling has Landau singularities and, if yes, whereprecisely these singularities are.

pQCD running couplings finite and monotonic in the infrared: when do they reflect the holomorphic properties of spacelike observables?