The b quark low-scale running mass m_kin
is determined from an analysis
of the Upsilon sum rules in the next-to-next-to-leading order (NNLO).
It is demonstrated that using this mass one can significantly
improve the convergence of the perturbation series for the
spectral density moments. We obtain
m_kin( 1 GeV ) = 4.56 \pm 0.06 GeV.
Using this result we derive the value of the MS-bar
mass m: 
m ) = 4.20 \pm 0.1 GeV.
Contrary to the low-scale running mass, the pole mass of the
b quark cannot be reliably determined from the sum rules.
As a byproduct of our study we find the NNLO analytical expression
for the cross section e+e- –> Q\bar Q of the quark antiquark
pair production in the threshold region, as well as the energy levels
and the wave functions at the origin for the ^1S_3
bound states of Q\bar Q.
