Vaughan identity
= Vaughan identity
{c}
{wiki=Vaughan's_identity}
For thresholds $U,V$, write $\mu_{\leq U}(d)=\mu(d)1_{d\leq U}$ and similarly for the other truncations. Vaughan's identity is
$$
\Lambda
=\Lambda_{\leq V}
+\mu_{\leq U}*\log
-\mu_{\leq U}*\Lambda_{\leq V}*1
+\mu_{>U}*\Lambda_{>V}*1.
$$
It decomposes a sum weighted by $\Lambda$ into short, Type I, and Type II bilinear sums.