  We compute both sides of the Witten-Veneziano formula using lattice
techniques. For the one side we perform dedicated quenched simulations and use
the spectral projector method to determine the topological susceptibility in
the pure Yang-Mills theory. The other side we determine in lattice QCD with
$N_f=2+1+1$ dynamical Wilson twisted mass fermions including for the first time
also the flavour singlet decay constant. The Witten-Veneziano formula
represents a leading order expression in the framework of chiral perturbation
theory and we also employ leading order chiral perturbation theory to relate
the flavor singlet decay constant to the relevant decay constant parameters in
the quark flavor basis and flavor non-singlet decay constants. After taking the
continuum and the SU$(2)$ chiral limits we compare both sides and find good
agreement within uncertainties.
