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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.

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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.

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