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

${\displaystyle L(s,\chi )=\sum _{n}{\frac {\chi (n)}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-\chi (p)p^{-s}}}}$

The Dirichlet L-function at negative integers is given by

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

where Bn is a generalized Bernoulli number defined by

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

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

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

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

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

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

## Viewed as a p-adic measure

p-adic L-functions can also be thought of as p-adic measures (or p-adic distributions) on p-profinite Galois groups. The translation between this point of view and the original point of view of Kubota–Leopoldt (as Qp-valued functions on Zp) is via the Mazur–Mellin transform (and class field theory).

## Totally real fields

Deligne & Ribet (1980), building upon previous work of Serre (1973), constructed analytic p-adic L-functions for totally real fields. Independently, Barsky (1978) and Cassou-Noguès (1979) did the same, but their approaches followed Takuro Shintani's approach to the study of the L-values.