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The analytic result for the singlet part of the Adler function of the vectorcurrent in a general gauge theory is presented in five-loop approximation.Comparing this result with the corresponding singlet part of theGross-Llewellyn Smith sum rule [1], we successfully demonstrate the validity ofthe generalized Crewther relation for the singlet part. This provides anon-trivial test of both our calculations and the generalized Crewtherrelation. Combining the result with the already available non-singlet part ofthe Adler function [2,3] we arrive at the complete ${\cal O}(\alpha_s^4)$expression for the Adler function and, as a direct consequence, at the complete${\cal O}(\alpha_s^4)$ correction to the $e^+ e^-$ annihilation into hadrons ina general gauge theory.

Adler Function, Sum Rules and Crewther Relation of Order O(alpha_s^4): the Singlet Case