# upper bound for an incomplete exponential sum

Let $q$ be a large prime and $\delta\in(0,1)$. Let $k$ be an integer which is not a multiple of $q$. Define $e(x)=e^{-2\pi ix}$. Can we get any non-trivial upper bounds for $$\sum_{a=1}^{q^{1-\delta}}e(\frac{k}{q}a)$$? A bound like $q^{1-\delta-\epsilon}$ would be good enough. I have searched literature in Weyl's sum and did not find any good bounds.

• As written, this is just a finite geometric series, which has an elementary closed form that can be used to estimate the size of the sum. Is that really the sum you mean? – Noam D. Elkies Mar 7 '14 at 23:32
• Yes. I have tried the geometric series formula and was unable to use that form to get a non-trivial upper bound in terms of a power of q. Maybe I need to take the magnitude of k into consideration. – Tony B Mar 8 '14 at 21:43

In general no non-trivial bound exists. Suppose e.g. that $a=1$. Then for all $k$ in the range of summation we have $\frac{ka}{q}\in[0, q^{-\delta}]$, thus $e(\frac{ka}{q})=1+\mathcal{O}(q^{-\delta})$.
If you have more information on $a$ you might use $$\sum_{a=1}^{N}e(\frac{ak}{q}) = \frac{1-e(\frac{(N+1)a}{q})}{1-e(\frac{k}{q})}\ll\frac{q}{a},$$ which is non-trivial for $a>q^\delta$.
• @Dong He means suppose $k=1$. – Will Sawin Apr 13 '14 at 22:34