I have the following exponential sum:

$\sum _{M<n\leq N}e\left (x/n^2\right )=\sum f(n),$

say, where $M$ and $N$ are something like $x^{1/4}$ and $x^{1/2}$.

My question is basically, how do I bound this?

I know of two methods used to bound exponential sums, the Weil way and the Van der Corput way. I think the first way is not really applicable here, but I would have guessed, based just on how it looks, that the Van der Corput method would have worked. However, that way would give, I believe, a bound of the type

$\frac {f'(N)+f'(M)}{(f''_{inf})^{1/2}}$

where $f''_{inf}\leq f''(n)$ for the whole range of summation.

This bound however is worse than trivial, so there's something happening there which I don't understand and which means the sum really is a different type of sum. Therefore, how should I bound it? There clearly is some oscillation, so it should be possible to say something. And also, as a curiosity question, why do we get absolutely no information on this sum using the Van der Corput method?

Thanks very much.