Unit function

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

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

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) (not to be confused with μ(n)).

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.