Unit function

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

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

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 ${\displaystyle u(n)}$ (not to be confused with ${\displaystyle \mu (n)}$, which generally denotes the Möbius function).

See also

• Möbius inversion formula
• Heaviside step function
• Kronecker delta

References

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.