### abstract

- The finite nth polylogarithm li n (z) ∈ ℤ/p(z) is defined as ∑k=1p−1z k /k n . We state and prove the following theorem. Let Li k : ℂ p → ℂ p be the p-adic polylogarithms defined by Coleman. Then a certain linear combination F n of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p1−nDF n (z) reduces modulo p>n+1 to lin−1(σ(z)), where D is the Cathelineau operator z(1−z)d/dz and σ is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.