Does anyone have any references for iterated exponential sums? That is, sums like


$$\sum\_{1\leq n\leq X} e(f(n))\sum\_{1\leq m\leq n} e(f(m)),$$


where $e(x)=e^{2\pi i x}$? I am looking for references that give bounds, limit theorems, anything like that.



Iterated exponential sums
This sum is equal to
$$
\frac{\left( \sum\_{n \leq x} e(f(n)) \right)^2 - \sum\_{n \leq x} e(2 f(n))}{2}
$$
In most situations, we do not expect there to be more than square root cancellation, and thus one would expect that this sum is asymptotically equal to
$$
\frac{1}{2} \left( \sum\_{n \leq x} e(f(n)) \right)^2
$$
(This is because the second sum is $\mathcal{O} (x)$ and we wouldn't expect the first sum to be less than $x \log \log x$ or something like that).


