# Is there an arithmetic function $\alpha(n)$ such that $\mu(n)=(\alpha*1)(n)$?

I've been looking into finding an arithmetic function $$\alpha(n)$$ for which its Dirichlet Convolution with the constant function $$1$$ is the Mobius Function, i.e.$$(\alpha(n)*1)=\mu(n)$$ I do not know if such a function exists, but any help is appreciated.

From here we can read that $$1 *\mu=\varepsilon$$ which is the identity of the Dirichlet convolution operation $$*$$. The identity of this operation, $$\varepsilon$$ is zero for all values in its domain except $$1$$ where $$\varepsilon(1)=1$$. So then multiplying by $$\mu$$ we have $$1*\mu*\mu=\epsilon*\mu =\mu$$. Then we should ask ourselves what is $$\mu*\mu$$? and the answer to this question is given here.