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.