= Vaughan identity proof
{c}
For <Dirichlet convolution>, let $\mathbf1(n)=1$ and $\epsilon(n)=1_{n=1}$. <Prime factorization> gives $\mu*\mathbf1=\epsilon$ and $\Lambda*\mathbf1=\log$. With subscripts denoting truncation at $U,V$,
$$
\begin{aligned}
\mu*\log&=\Lambda,\\
\mu_{>U}*\mathbf1*\Lambda_{>V}
&=(\epsilon-\mu_{\leq U}*\mathbf1)*\Lambda_{>V}\\
&=\Lambda_{>V}-\mu_{\leq U}*\log+\mu_{\leq U}*\mathbf1*\Lambda_{\leq V}.
\end{aligned}
$$
Rearranging proves the <Vaughan identity>.
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