# Bound on exponential sum with weights

Let $e(z)$ denote $e^{2 \pi i z}$ and let $f(z)$ a smooth real function. I know one can bound sums of the form $$\sum_{x \leq X} e(f(x))$$ via for example Van der Corputs's result, provided we make assumptions on the range of the derivatives. I was wondering if there were results of this type for weighted sums. Can we bound a sum of the form $$\sum_{x \leq X} g(x) e(f(x)) ?$$ If so I was wondering what kind of assumption would be needed for g(z). (In particular, would it possible for $g(z) = 2^z$, for example? ) Thanks!

• if $g(z)=2^z$, then the last summand should give the dominant contribution to the sum. If $g$ varies slowly, then you can use partial summation. Look also in Titchmarsch and in Iwaniec-Kowalski for various results about exponential sums. – Dorian Coughlan Nov 19 '14 at 17:21