 Identity for Dirichlet convolution

The Identity for Dirichlet convolution is the arithmetic functionFor every arithmetic function $f$, it satisfies $f\ast \epsilon =\epsilon \ast f=f$. Indeed, in $(f\ast \epsilon )(n)={\sum }_{d\mid n}f(d)\epsilon (n/d)$, only $d=n$ survives. For example,At $n=1$ the sole term is $f(1)\epsilon (1)=f(1)$. This gives a useful way to check Dirichlet convolution calculations: inserting $\epsilon$ leaves every value unchanged. The three arithmetic functions $1$, $\mathrm{id}$, and $\epsilon$ all take value $1$ at $1$, but at $12$ the constant-one arithmetic function, the identity arithmetic function, and the Identity for Dirichlet convolution take values $1$, $12$, and $0$, respectively. ^[ref]

The Möbius function is the inverse of the constant-one arithmetic function under Dirichlet convolution, meaning $\mu \ast 1=1\ast \mu =\epsilon$. The Möbius divisor-sum identity gives a short proof: for $n>1$ the nonzero terms expand ${\prod }_{p\mid n}(1-1)=0$, while at $1$ the sum is $\mu (1)=1$. At $12$, the computation isThe zeros occur because $4$ and $12$ are divisible by the square of the prime number $2$. Consequently, if $F=f\ast 1$ is a divisor sum, then associativity gives $F\ast \mu =f\ast (1\ast \mu )=f\ast \epsilon =f$: this is Möbius inversion.