  The SIDIS transverse spin asymmetries weighted with powers of $P_T$, the
hadron transverse momentum in the $\gamma N$ reference system,have been
introduced already twenty years ago and are considered quite interesting. While
the amplitudes of the modulations in the azimuthal distribution of the hadrons
are the ratios of convolutions over transverse momenta of the
transverse--momentum dependent parton distributions and of the corresponding
fragmentation functions, and can be evaluated analytically only making
assumptions on the transverse--momentum dependence of these functions, the
weighted asymmetries allow to solve the convolution integrals over
transverse--momenta without those assumptions. Using the high statistics data
collected in 2010 on transversely polarized proton target COMPASS has evaluated
in x-bins the $P_T$ weighted Sivers asymmetry which is 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 asymmetry.
