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(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.Comment: 22 pages, Late