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
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.