# Unit function

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In [number theory](/source/number_theory), the '''unit function''' is a completely [multiplicative function](/source/multiplicative_function) on the positive integers defined as:

:<math>\varepsilon(n) = \begin{cases} 1, & \mbox{if }n=1 \\ 0, & \mbox{if }n \neq 1 \end{cases} </math>

It is called the unit function because it is the [identity element](/source/identity_element) for [Dirichlet convolution](/source/Dirichlet_convolution).<ref>{{citation
 | last = Estrada | first = Ricardo
 | doi = 10.1216/jiea/1181075867
 | issue = 2
 | journal = Journal of Integral Equations and Applications
 | mr = 1355233
 | pages = 159–166
 | title = Dirichlet convolution inverses and solution of integral equations
 | volume = 7
 | year = 1995| doi-access = free
 }}.</ref>

It may be described as the "[indicator function](/source/indicator_function) of 1" within the set of positive integers.  It is also written as <math>u(n)</math> (not to be confused with <math>\mu(n)</math>, which generally denotes the [Möbius function](/source/M%C3%B6bius_function)).

==See also==
* [Möbius inversion formula](/source/M%C3%B6bius_inversion_formula)
* [Heaviside step function](/source/Heaviside_step_function)
* [Kronecker delta](/source/Kronecker_delta)

==References==
{{reflist}}

Category:Multiplicative functions
Category:1 (number)

{{numtheory-stub}}

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Adapted from the Wikipedia article [Unit function](https://en.wikipedia.org/wiki/Unit_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Unit_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.