Electron-positron annihilation to photons at $O(\alpha^3)$ revisited.

We apply the modern multiloop methods to the calculation of the total cross sections of electron-positron annihilation to 2 and 3 photons exactly in $s/m^2$ with the accuracy $O(\alpha^3)$. Examining the asymptotics of our results, we find agreement with Ref. [Andreassi_et_al_1962] and discover mistakes in the results of Refs. [Eidelman&Kuraev_1978,Berends&Kleiss_1981]. This mistake is due to the terms, omitted in differential cross section in Refs. \[Eidelman&Kuraev_1978,Berends&Kleiss_1981], which are peaked in the kinematic region with all three photons being quasi-parallel to the collision axis. After restoring these terms, we find an agreement of the corrected result of Ref. [Berends&Kleiss_1981] with our result.