Next-to-next-to-leading order vacuum polarization function of heavy quark near threshold and sum rules for $b \bar b$ system

A correlator of the vector current of a heavy quark is computed analytically near threshold in the next-to-next-to-leading order in perturbative and relativistic expansion that includes $\al_s^2$, $\al_sv$ and $v^2$ corrections in the coupling constant and velocity of the heavy quark to the nonrelativistic Coulomb approximation. Based on this result, the numerical values of the $b$-quark pole mass and the strong coupling constant are determined from the analysis of sum rules for the $\Upsilon$ system. The next-to-next-to-leading corrections are found to be of order of next-to-leading ones.