On Coleman Integration and p-adic L-functions

In this article we discuss the integration theory on p-adic projective space of dimension 1, and apply it to construct the logarithmic F-crystal on the p-adic complete field, where polylogarithm functions occurs in a natural development. The usage of polylogarithms realize in the computation for the p-adic L-values. To be precise, valuation of the polylogarithms at primitive roots of unity is related to the special values of the Kubota-Leopold L-function at positive integers. Eventually, we conclude by deriving a formula relating the evaluation of p-adic L-functions at k to the k-th polylogarithm, which extends the formula by Koblitz, who proved the case k=1.