In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic family of Galois representations, and the image could be the p-adic numbers Qp or its algebraic closure.

Dirichlet L-functions

The Dirichlet L-function is given by the analytic continuation of

$L(s,\chi) = \sum_n\frac{\chi(n)}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-\chi(p)p^{-s}}$

The Dirichlet L-function at negative integers is given by

$L(1-n, \chi) = -\frac{B_{n,\chi}}{n}$

where Bn is a generalized Bernoulli number defined by

$\displaystyle \sum_{n=0}^\infty B_{n,\chi}\frac{t^n}{n!} = \sum_{a=1}^f\frac{\chi(a)te^{at}}{e^{ft}-1}$

for χ a Dirichlet character with conductor f.

Definition using interpolation

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

$\displaystyle L_p(1-n, \chi) = (1-\chi(p)p^{n-1})L(1-n, \chi)$

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

$\displaystyle L_p(1-n, \chi) = (1-\chi\omega^{-n}(p)p^{n-1})L(1-n, \chi\omega^{-n})$

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