# Definition:Thomae Function

(Redirected from Definition:Modified Dirichlet Function)

Jump to navigation
Jump to search
## Definition

The **Thomae function** $D_M: \R \to \R$ is the real function defined as:

- $\forall x \in \R: \map {D_M} x = \begin {cases} 0 & : x = 0 \text { or } x \notin \Q \\ \dfrac 1 q & : x = \dfrac p q : p \perp q, q > 0 \end {cases}$

where:

- $\Q$ denotes the set of rational numbers
- $p \perp q$ denotes that $p$ and $q$ are coprime (that is, $x$ is a rational number expressed in canonical form)

## Also known as

The **Thomae function** is also seen styled as **Thomae's Function**.

It has several names in the literature:

- the
**modified Dirichlet function**after Johann Peter Gustav Lejeune Dirichlet - the
**(small) Riemann function**after Georg Friedrich Bernhard Riemann - the
**popcorn function** - the
**raindrop function** - the
**countable cloud function** - the
**ruler function**(although strictly speaking the ruler function is a restriction of this to the dyadic rationals) **Stars over Babylon**(coined by John Horton Conway)

## Also see

- Results about
**the Thomae function**can be found**here**.

## Source of Name

This entry was named for Carl Johannes Thomae.

## Sources

- Weisstein, Eric W. "Dirichlet Function." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletFunction.html