A lot of exponential equations in Lie algebras remain unsolved if they require to expand the expression log(exp(X)exp(Y)) for non-commuting variables X and Y. The classical Baker-Campbell-Hausdorff formula (BCH) gives only a recursive way to compute H=log(exp(X)exp(Y)) via commutators of X and Y. The Dynkin form of BCH is also awkward to use because it contains infinitely many similar terms. The series H lives in the free Lie algebra L generated by X,Y.

The first aim of the talk is to present a compressed version of Campbell-Hausdorff formula in the quotient L/[[L,L],[L,L]]. The second aim is to apply the compressed formula to describe explicitly all Drinfeld associators in the same quotient. Originally, Drinfeld associators appeared in the theory of quasi-Hopf algebras. Formally, an associator is a solution of complicated algebraic equations (pentagon and hexagon) involving 5 and 6 exponentials. The compressed formula is a powerful tool that helps to solve the above exponential equations completely.

In the context of number theory the main result states that the pentagon and hexagon equations do not provide polynomial relations between odd values of the classizal zeta function. The talk is based on the article "Compressed Drinfeld associators" recently published in J. of Algebra 292 (2005), 184-242.