Basic question: say you have a sum $$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$ where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k) = g(n_1)\dotsb g(n_k)$, etc.). Obviously we'd like to make the ranges of $n_1,\dotsc,n_k$ independent - letting each $n_i$ take values in a range $[y_i, c y_i]$, say. I would like to see some different ways to go about it.
(I've edited this question: I originally asked "what is your favorite way to go about it?", but was told that was too subjective. At any rate: any analytic number theorist has a way to do the above; I would simply like to see what different people do, with a special interest in ways that have the merit of simplicity or elegance, and keep that merit when used to prove actual bounds, perhaps even explicit bounds, as opposed to just showing that something tends to zero when divided by the trivial bound. I know "simplicity" and "elegance" are subjective, but one would think that a simple or elegant procedure can just be described objectively.)
Personally, I think it is nice to use the equality $$\int_{1/c}^\infty 1_{(t,ct]}(x) \frac{dt}{t} = \log c,$$ valid for $x\geq 1$. Then we can write $$\frac{1}{(\log c)^k} \int_{\substack{t_1,\dotsc,t_k\geq 1/c\\ t_1 t_2 \dotsc t_k \leq x}} \sum_{t_1<n_1\leq c t_1} \dotsc \sum_{t_k<n_k\leq c t_k} f(n_1,\dotsc,n_k) = \sum_{n_1,\dotsc,n_k} f(n_1,\dotsc,n_k) \eta\left(\frac{n_1 n_2 \dotsb n_k}{x}\right),$$ where $\eta$ is a smoothing such that $\eta(t)=1$ for $t\leq 1$, $0<\eta(t)<1$ for $1<t<c^k$ and $\eta(t)=0$ for $t\geq c^k$. Then one can either let $c\to 1^+$ or just live with the smoothing $\eta$.
Either is a good option when $k$ is bounded. But what when $k$ is unbounded? Both options seem a bit nastier.
