Finite Polylogarithms, Their Multiple Analogues and the Shannon Entropy

We show that the entropy function—and hence the finite 1-logarithm—behaves a lot like certain derivations. We recall its cohomological interpretation as a 2-cocycle and also deduce 2n-cocycles for any n. Finally, we give some identities for finite multiple polylogarithms together with number theoretic applications. 1 Information theory, Entropy and Polylogarithms It is well known that the notion of entropy occurs in many sciences. In thermodynamics, it means a measure of the quantity of disorder, or more accurately, the tendancy of a system to go toward a disordered state. In information theory, the entropy measures (in terms of real positive numbers) the quantity of information of a certain property [17],[20]. From a practical viewpoint, entropies play also a key role in the study of random bit generators (deterministic or not) [8], in particular due to the Maurer test [16]. A general definition of entropy has been given by Rényi [18]: let S = {s1, . . . , sn} be a set of discrete events for which the probabilities are given by pi = P (s = si) for i = 1, . . . , n. The Rényi entropy S is then defined for α > 0 and α 6= 1 as Hα(S) = 1 1− α log ( n ∑