Unit function

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

${\displaystyle \varepsilon (n)={\begin{cases}1,&{\mbox{if }}n=1\0,円&{\mbox{if }}n\neq 1\end{cases}}}${\displaystyle \varepsilon (n)={\begin{cases}1,&{\mbox{if }}n=1\0,円&{\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 ${\displaystyle u(n)}${\displaystyle u(n)} (not to be confused with ${\displaystyle \mu (n)}${\displaystyle \mu (n)}, which generally denotes the Möbius function).

• Möbius inversion formula
• Heaviside step function
• Kronecker delta

• Multiplicative functions
• 1 (number)
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