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Von Mangoldt function

{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.


Vaughan identity

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 3 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 117 / 3 / f / Solution

Vaughan identity

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Vaughan identity

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