= Vaughan identity for the Möbius function
{c}
{title2=$\mu=\mu_{\leq U}+\mu_{\leq V}-\mu_{\leq U}*\mu_{\leq V}*\mathbf1+\mu_{>U}*\mu_{>V}*\mathbf1$}
The displayed identity is exact for all positive integers and all positive cutoffs. Expand $(\mu-\mu_{\leq U})*(\mu-\mu_{\leq V})*\mathbf1$ and use $\mu*\mathbf1=\varepsilon$. The two truncated terms are short sums; the negative term is a <type I sum> over <divisors> at most $UV$; the last term is a <type II sum> with both factor variables larger than a cutoff. This form of the <Vaughan identity> is specific to Möbius weights.
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