In algebra and number theory, a **distribution** is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function.

## Contents

- Hurwitz zeta function
- Bernoulli distribution
- Cyclotomic units
- Universal distribution
- Stickelberger distributions
- p adic measures
- Hecke operators and measures
- References

The original examples of distributions occur, unnamed, as functions φ on **Q**/**Z** satisfying

We shall call these **ordinary distributions**. They also occur in *p*-adic integration theory in Iwasawa theory.

Let ... → *X*_{n+1} → *X*_{n} → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let *X* be their projective limit. We give each *X*_{n} the discrete topology, so that *X* is compact. Let φ = (φ_{n}) be a family of functions on *X*_{n} taking values in an abelian group *V* and compatible with the projective system:

for some *weight function* *w*. The family φ is then a *distribution* on the projective system *X*.

A function *f* on *X* is "locally constant", or a "step function" if it factors through some *X*_{n}. We can define an integral of a step function against φ as

The definition extends to more general projective systems, such as those indexed by the positive integers ordered by divisibility. As an important special case consider the projective system **Z**/*n***Z** indexed by positive integers ordered by divisibility. We identify this with the system (1/*n*)**Z**/**Z** with limit **Q**/**Z**.

For *x* in *R* we let ⟨*x*⟩ denote the fractional part of *x* normalised to 0 ≤ ⟨*x*⟩ < 1, and let {*x*} denote the fractional part normalised to 0 < {*x*} ≤ 1.

## Hurwitz zeta function

The multiplication theorem for the Hurwitz zeta function

gives a distribution relation

Hence for given *s*, the map
**Q**/**Z**.

## Bernoulli distribution

Recall that the *Bernoulli polynomials* *B*_{n} are defined by

for *n* ≥ 0, where *b*_{k} are the Bernoulli numbers, with generating function

They satisfy the *distribution relation*

Thus the map

defined by

is a distribution.

## Cyclotomic units

The cyclotomic units satisfy *distribution relations*. Let *a* be an element of **Q**/**Z** prime to *p* and let *g*_{a} denote exp(2πi*a*)−1. Then for *a*≠ 0 we have

## Universal distribution

One considers the distributions on *Z* with values in some abelian group *V* and seek the "universal" or most general distribution possible.

## Stickelberger distributions

Let *h* be an ordinary distribution on **Q**/**Z** taking values in a field *F*. Let *G*(*N*) denote the multiplicative group of **Z**/*N***Z**, and for any function *f* on *G*(*N*) we extend *f* to a function on **Z**/*N***Z** by taking *f* to be zero off *G*(*N*). Define an element of the group algebra *F*[*G*(*N*)] by

The group algebras form a projective system with limit *X*. Then the functions *g*_{N} form a distribution on **Q**/**Z** with values in *X*, the **Stickelberger distribution** associated with *h*.

## p-adic measures

Consider the special case when the value group *V* of a distribution φ on *X* takes values in a local field *K*, finite over **Q**_{p}, or more generally, in a finite-dimensional *p*-adic Banach space *W* over *K*, with valuation |·|. We call φ a **measure** if |φ| is bounded on compact open subsets of *X*. Let *D* be the ring of integers of *K* and *L* a lattice in *W*, that is, a free *D*-submodule of *W* with *K*⊗*L* = *W*. Up to scaling a measure may be taken to have values in *L*.

## Hecke operators and measures

Let *D* be a fixed integer prime to *p* and consider **Z**_{D}, the limit of the system **Z**/*p*^{n}*D*. Consider any eigenfunction of the Hecke operator *T*^{p} with eigenvalue *λ*_{p} prime to *p*. We describe a procedure for deriving a measure of **Z**_{D}.

Fix an integer *N* prime to *p* and to *D*. Let *F* be the *D*-module of all functions on rational numbers with denominator coprime to *N*. For any prime *l* not dividing *N* we define the *Hecke operator* *T*_{l} by

Let *f* be an eigenfunction for *T*_{p} with eigenvalue λ_{p} in *D*. The quadratic equation *X*^{2} − λ_{p}*X* + *p* = 0 has roots π_{1}, π_{2} with π_{1} a unit and π_{2} divisible by *p*. Define a sequence *a*_{0} = 2, *a*_{1} = π_{1}+π_{2} = *λ*_{p} and

so that