I consider a gas of free'' anyons with statistical parameter {delta}, with hard cores, on a two-dimensional square lattice. Using a recently derived Jordan-Wigner transformation, I map this problem onto a gas of fermions on the same lattice coupled to a Chern-Simons gauge theory with coupling {theta}=1/2{delta}. At the semiclassical level, the system is found to be equivalent to a gas of fermions, with the same density, in an average effective magnetic field {rho}/{theta}. I consider the case in which an integer number of the Landau bands of the saddle-point problem are completely filled. If {delta}={pi}/{ital m} and the density {rho}={ital r}/{ital q}, with {ital m}, {ital r}, and {ital q} integers, the system is a superfluid, provided that {ital q} is larger than twice the largest common factor of {ital m} and {ital r}. If {ital q} is even and the system is half filled, the state may be either a superfluid or a quantum Hall phase. For all other values of {rho} and {delta}, compatible with integer filling of the Landau bands, the system is in a quantum Hall phase. The dynamical stability of the superfluid state is ensured by the topological invariance of the quantized Hall conductancemore » of the fermion problem. I find a close analogy between anyon superconductivity and the Schwinger mechanism. The effective Lagrangian for the low-energy modes coupled to the electromagnetic field is derived. The energies of fermion and flux states are logarithmically divergent, but finite for the anyon state. The system has flux quantization, a zero-temperature Hall effect with a quantized Hall conductance, Meissner effect, charged vortices, screening with induced magnetic fields for static charges, and different masses for the longitudinal and transverse components of the electromagnetic field.« less