Distributions of Manin’s iterated integrals

We recall the definition of Manin’s iterated integrals of a given length. We then explain how these generalise modular symbols and certain aspects of the theory of multiple zeta-values. In length one and two we determine the limiting distribution of these iterated integrals. Maybe surprisingly, even if we can compute all moments also in higher length we cannot in general determine a distribution for length three or higher. This is joint work with N. Matthes.