I know that Vaughan's identity is one of the methods used in analityc number theory, I would like see an example, I say a simple example of application of this theorem for encourage to study the statement of such theorem, and perhaps its proof. I say this cited theorem, Vaughan's identity, concerning
$$\sum_{n\leq N}\Lambda(n)f(n),$$ where $\Lambda(n)$ is the von Mangoldt function, and for example (you can choose other function $f(x)$ for this didactic example)
$$f(n) = \begin{cases} e^{2\pi ig(n)}, & \text{if $1\leq n\leq N$} \\ 0, & \text{otherwise} \end{cases}$$ for an easy function $g(n)$. I hope that this last has sense for you. You can choose other functions to show an easy application of Vaughan's identity.
Question. Can you give a didactic example of an application, for a choice of $g(n)$ or other $f(x)$, of cited theorem? I would like to see easy computations for previous identity in RHS, if you can use Vinogradov notation $\ll$ in some computations of such summands in RHS, for illustrate the application of the theorem and how you deduce some easy computations, you are welcome. Thanks in advance.
Here is a first example for $f(n)$, namely $f(n)=n^{-s}$ for complex $s$ with $Re(s)>1$. Then we obtain the partial sums of the Dirichlet series $$ \sum_{n=1}^{\infty}\Lambda(n)n^{-s}=-\frac{\zeta'(s)}{\zeta(s)}, $$ involving $\zeta(s)$ and its derivative. For more substantial applications of Vaughan's identity see here.