  The transverse spin asymmetries measured in semi-inclusive leptoproduction of
hadrons, when weighted with the hadron transverse momentum $P_T$, allow for the
extraction of important transverse-momentum-dependent 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
$P_T$-weighted Sivers asymmetries, which are both proportional to the product
of the firsttransverse 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.
