Find bound on the following sums for $k\in \{1,\dots p-1\}$ where $p$ is a prime (even assumed to be $3\mod 4$, I used this to get to the current sum) find good upper and lower bounds on
$$\sum_{s=0}^{\sqrt{p}}1-\cos\left(\dfrac{2\pi }{p}ks^{2} \right)$$
Here I tried to use a Taylor expansion of various degrees and I end up having to sum over consecutive integers power of order $4,8$ and I was wondering was there another simple ways of getting bounds?
If it's just a finite sum over consecutive integer powers of 4 or 8, then there is a closed form and wolfram can definitely help. Sorry to comment in the answer, my account is not rich enough to comment yet.