Assessing signals of TMD physics in SIDIS azimuthal asymmetries and in the extraction of the Sivers function

New data on the Sivers azimuthal asymmetry measured in semi-inclusive deep- inelastic scattering processes have recently been released by the COMPASS Collaboration at CERN. Their increased precision and their particular binning, in terms of Q^2 as well as x, motivates a new extraction of the Sivers function, within the framework of a simple and transparent parametrization. Signals of TMD effects visible in the Sivers asymmetries are critically assessed. A thorough study of the uncertainties affecting the extracted Sivers function is presented, including the low-x and large-x regions.