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

With ${\mu }_{\le U}(d)=\mu (d)1_{d\le U}$ and analogous notation for ${\Lambda }_{\le V}$, Vaughan identity isThus, when $UV\le N$,The first term is short, the next two are Type I sums, and the last becomes a Type II bilinear sum after grouping variables and applying a dyadic decomposition.