Unit function

In number theory, the unit function is a completely multiplicative function on the positive integers defined as: ε ( n ) = { 1 , if n = 1 0 , if n ≠ 1 {\displaystyle \varepsilon (n)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n\neq 1\end{cases}}} It is called the unit function because it is the identity element for Dirichlet convolution. It may be described as the "indicator function of 1" within the set of positive integers. It is also written as u ( n ) {\displaystyle u(n)} (not to be confused with μ ( n ) {\displaystyle \mu (n)} , which generally denotes the Möbius function).

In number theory, the unit function is a completely multiplicative function on the positive integers defined as:

ε ( n ) = { 1 , if  n = 1 0 , if  n ≠ 1 {\displaystyle \varepsilon (n)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n\neq 1\end{cases}}}

It is called the unit function because it is the identity element for Dirichlet convolution.^[1]

It may be described as the "indicator function of 1" within the set of positive integers. It is also written as u ( n ) {\displaystyle u(n)} (not to be confused with μ ( n ) {\displaystyle \mu (n)} , which generally denotes the Möbius function).

See also

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• Möbius inversion formula
• Heaviside step function
• Kronecker delta

References

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1. ^ Estrada, Ricardo (1995), "Dirichlet convolution inverses and solution of integral equations", Journal of Integral Equations and Applications, 7 (2): 159–166, doi:10.1216/jiea/1181075867, MR 1355233.

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