Let \[ S_{n,m}(q)=\sum_{a=1\atop {(a,q)=1}}^qe\left (\frac {an+\overline am}{q}\right )\] be Kloosterman's sum and $\alpha _n,\beta _m$ be complex numbe of modulus $\leq 1$. For $Q,N,M>0$ what is the best bound for \[ S:=\sum_{n\sim N}\alpha_n\sum_{m\sim M}\beta_m\sum_{q\sim Q}S_{n,m}(q)?\]
I'm finding it difficult to find some kind of summary of what's known. I know (and obviously correct me if I'm wrong, or missing something important) that
up to gcd's and $\epsilon $'s, we have $S_{n,m}(q)\ll \sqrt q$, so a trivial bound is essentially $S\ll NMQ^{3/2}$.
It is expected that $S_{n,m}(q)$ is on average $\ll 1$ when averaging smoothly over $q$, and I can find stuff on this in Chapter 16 of Iwaniec-Kowalski. With no smoothing, it is expected to still be true, but something corresponding to $S_{n,m}(q)\ll _{n,m}q^{1/6}$ is all we know.
A paper of Sarnak Tsimerman shows we can, at least for $N,M$ not too large, remove the $n,m$ dependence in the $\ll _{n,m}$ above, so we have $S\ll NMQ^{7/6}$.
An important paper was one of Deshouillers and Iwaniec from the 80's which showed that $S\ll NMQ$ (with smoothing).
However I don't really know what to expect when considering possible cancellation over $n,m$ too. For example, in papers like this one https://www.epfl.ch/labs/tan/wp-content/uploads/2018/10/bilinearforms.pdf it seems (if I'm not misunderstanding anything) the main point is getting cancellation when just averaging over $n,m$. But maybe that question is irrelevant when averaging over $q$ too? Is it possible to get something better than $S\ll NMQ$ for example?
As you can see, I don't have the right of idea of what bounds are possible and what to expect when averaging over $n,m,q$, and any useful "think about it like this"-s would be appreciated. Or any papers which have a particularly lucid summary of bounds in various ranges.
