Let $\omega (n)$ be the number of (distinct) prime divisors of $n$ $$\omega (n)=\sum _{p|n}1$$ and let $S(a/q)$ be its exponential sum $$\sum _{n\leq x}\omega (n)e(na/q).$$
Question 1: Can anyone give me any references where this exponential sum was studied? (In fact I can't find any papers at all, which I guess is an ominous sign...)
I want to take $q$ up to $\sqrt x$ and I would like a bound $$\sideset {}{'}\sum _{a=1}^qe(\overline a/q)S(a/q)\ll x\hspace {10mm}(1)$$ (or be told, for $q>1$, that it can't be true...). Without the oscillatory $e(\overline a/q)$ factor the sum would become $$\sum _{n\leq x}\omega (n)c_q(n)\hspace {10mm}(2)$$ which I think I could study in the usual way using Perron's formula (though with a much less workable Dirichlet series). In this case I'd get an error of acceptable size (but also a main term), I think something like $q^{1-\theta }x^\theta $, where $\theta $ is the $1-1/\log T$ zero-free region for $\zeta (s)$.
So Question 2: Is (1) a much harder problem than (2)? (Is it hopeless?)
And finally one more question: Actually, in (1) I will sum over $q$ as well (and I have no absolute value signs). So if I sum over $a$ first, then I actually want to estimate $$\sum _{n\leq x}\omega (n)\sum _{q\leq Q}S(n,1;q)$$ where $$S(n,m;q)=\sideset {}{'}\sum _{a=1}^qe\left (\frac {na+m\overline a}{q}\right )$$ is Kloosterman's sum. I know there were major results in the 80's about evaluating sums of Kloosterman sums, and that $$\sum _{n,m,q}\alpha _n\beta _mS(n,m;q)$$ can be shown to have full cancellation $\ll (NMQ)^{1+\epsilon }$. In those cases $\alpha _n,\beta _m$ are general coefficients - is it ever possible to say more (as in, get an asymptotic, or at least get no $\epsilon $'s) if we have specific functions in mind? So final question I guess is something like when can I evaluate a sum of Kloosterman sums up to the right order of magnitude (averaging over all variables)? (I find a flurry of papers about results on averaging over just $n,m$, but my problem should be easier, being allowed to average over $q$?)
By the way, you can assume everything has smooth weights (the $n$ sum and the $q$ sum).
One example which I think helps me in understanding is to replace $\omega (n)$ with $d_k(n)$. In that case I can evaluate $S(a/q)$ using Voronoi's formula, and the $a\mapsto \overline a$ flip makes a lot of things simplify (in particular Kloosterman sums disappear). But for $\omega (n)$ I'm not going to have anything like this summation formula, so I'm not sure if I'm still actually pretty far away with that example.
Thanks for reading!
