Timelike and spacelike QCD characteristics of the $e^+e^-$ annihilation   process
  We consider the Adler $D$-function, which is defined in the spacelike region, and the $e^+e^-$ annihilation ratio smeared, according to the Poggio, Quinn, and Weinberg method, which is determined for timelike argument. We show that the method of the nonperturbative $a$-expansion allows one to describe these Euclidean and Minkowskian characteristics of the process of $e^+e^-$ annihilation down to the lowest energy scales. 
K.A. Milton, I.L. Solovtsov, and O.P. Solovtsova