## Zagier's conjecture on zeta(F,4)

Abstract: We prove that the weight 4 Beilinson's regulator map can be expressed via the classical n-logarithms, $n \leq 4$.

This plus Borel's theorem implies Zagier's conjecture, relating the value of the Dedekind zeta functions at $s=4$ and the classical tetralogarithm. Another application is to the values of L-functions of elliptic curves over $Q$ at $s=4$.

One of the new tools is a connection between cluster varieties and polylogarithms, generalising our work with V. Fock relating cluster varieties and the dilogarithm.