SUM EXPRESSIONS FOR KUBOTA–LEOPOLDT P-ADIC L-FUNCTIONS

摘要

When p is an odd prime, Delbourgo observed that any Kubota–Leopoldt p-adic L-function, when multiplied by an auxiliary Euler factor, can be written as an infinite sum. We shall establish such expressions without restriction on p, and without the Euler factor when the character is non-trivial, by computing the periods of appropriate measures. As an application, we will reprove the Ferrero–Greenberg formula for the derivative L'_p(0, χ). We will also discuss the convergence of sum expressions in terms of elementary p-adic analysis, as well as their relation to Stickelberger elements; such discussions in turn give alternative proofs of the validity of sum expressions.