Identity arithmetic function ($\mathrm{id}$)

The identity arithmetic function is $\mathrm{id}(n)=n$ on the positive integers, for example $\mathrm{id}(1)=1$ and $\mathrm{id}(12)=12$. It is a completely multiplicative function and gives the sum-of-divisors function through $\mathrm{id}\ast 1={\sigma }_1$ under Dirichlet convolution. It is distinct from the Identity for Dirichlet convolution $\epsilon$. The identity $\mathrm{id}=1\ast \phi$, where $\phi$ is Euler's totient function, is equivalent to ${\sum }_{d\mid n}\phi (d)=n$.