Fundamentals of p-adic multiple L-functions and evaluation of their special values

Abstract

We construct p-adic multiple L-functions in several variables, which are generalizations of the classical Kubota–Leopoldt p-adic L-functions, by using a specific p-adic measure. Our construction is from the p-adic analytic side of view, and we establish various fundamental properties of these functions. (a) Evaluation at non-positive integers: We establish their intimate connection with the complex multiple zeta-functions by showing that the special values of the p-adic multiple L-functions at non-positive integers are expressed by the twisted multiple Bernoulli numbers, which are the special values of the complex multiple zeta-functions at non-positive integers. (b) Multiple Kummer congruences: We extend Kummer congruences for Bernoulli numbers to congruences for the twisted multiple Bernoulli numbers. (c) Functional relations with a parity condition: We extend the vanishing property of the Kubota–Leopoldt p-adic L-functions with odd characters to our p-adic multiple L-functions. (d) Evaluation at positive integers: We establish their close relationship with the p-adic twisted multiple star polylogarithms by showing that the special values of the p-adic multiple L-functions at positive integers are described by those of the p-adic twisted multiple star polylogarithms at roots of unity, which generalizes the result of Coleman in the single variable case.