P-adic L-function - Definition Using Interpolation

Definition Using Interpolation

The Kubota–Leopoldt p-adic L-function L_p(s, χ) interpolates the Dirichlet L-function with the Euler factor at p removed. More precisely, L_p(s, χ) is the unique continuous function of the p-adic number s such that

for positive integers n divisible by p − 1. The right hand side is just the usual Dirichlet L-function, except that the Euler factor at p is removed, otherwise it would not be p-adically continuous. The continuity of the right hand side is closely related to the Kummer congruences.

When n is not divisible by p − 1 this does not usually hold; instead

for positive integers n. Here χ is twisted by a power of the Teichmuller character ω.