I am reading Melvyn Nathanson's book Additive Number Theory - Classical bases, and in particular getting through the proof of Vinogradov's theorem (also known as the ternary Goldbach theorem). Here the proof is due to Vaughan. In particular, to get a good bound on the contribution of the minor arc, one estimates the absolute value of $F(\alpha) = \displaystyle \sum_{p \leq N} \log(p) e(p \alpha)$ by breaking it up into three pieces via something called Vaughan's Lemma. The result is as follows:

For $u \geq 1$, let $M_u(k) = \displaystyle \sum_{d | k, d \leq u} \mu(d)$. Let $\Phi(k,l)$ be an arithmetic function of two variables. Then $\displaystyle_{u < l \leq N} \Phi(1,l) + \sum_{u < k \leq N} \sum_{u < l \leq N/k} M_u(k)\Phi(k,l) = \sum_{d \leq u} \sum_{u < l \leq N/d} \sum_{m \leq N/ld} \mu(d)\Phi(dm,l)$

I am wondering if there are other applications of this lemma. In the proof of Vinogradov's theorem, we set $u = N^{2/5}$and $\Phi(k, l) = \Lambda(l)e(\alpha k l)$ and obtain estimates on each of the pieces. In my current problem I need to improve on these estimates slightly, so I want to see how flexible Vaughan's lemma can be by looking at cases where it is used to solve other problems. Any suggestions would be welcome.