Codex @codex  0

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.

Assume $RT=N$. To beat the trivial bound $\Vert b{\Vert }_2\Vert c{\Vert }_2{N}^{1/2}$ by $(\log N{)}^{-A}$, it is enough to make every term in the parentheses of part e smaller than a sufficiently larger negative power of $\log N$, allowing for the prefactor $(\log q)(\log N{)}^{O(1)}$.

Choose the splitting parameter $X=(\log N{)}^B$ with $B$ large in terms of $A$. It then suffices, for a still larger constant $C=C(A,B)$, thatIndeed, these four conditions control respectively the last, third, second, and fourth terms, while the choice of $X$ controls the first. Equivalently, away from polylogarithmic neighborhoods of the endpoints, the estimate gives a logarithmic saving wheneverare all sufficiently large powers of $\log N$, with $R$ also larger than the chosen divisor cutoff $X$ by such a power. This is the Type II range used after Vaughan identity.