The transverse spin asymmetries measured in semi-inclusive leptoproduction of hadrons, when weighted
with the hadron transverse momentum PT , allow for the extraction of important transverse-momentumdependent distribution functions. In particular, the weighted Sivers asymmetries provide direct information
on the Sivers function, which is a leading-twist distribution that arises from a correlation between the transverse momentum of an unpolarised quark in a transversely polarised nucleon and the spin of the nucleon.
Using the high-statistics data collected by the COMPASS Collaboration in 2010 with a transversely polarised proton target, we have evaluated two types of PT -weighted Sivers asymmetries, which are both
proportional to the product of the first transverse moment of the Sivers function and of the fragmentation
function. The results are compared to the standard unweighted Sivers asymmetries and used to extract the
first transverse moments of the Sivers distributions for u and d quarks
