Make Weyl-van der Corput estimates explicit:

Let $f:[a,b]\rightarrow\mathbb{R}$ be a somewhat smooth function, and assume that you have some bounds on certain derivatives of $f$. Then give an upper bound for $\sum_{n=N}^{2N} e^{2\pi i f(n)}$.

There are deep qualitative bounds, however, not much work has been done on the constants involved in these bounds. While it is trivial to get some constants, getting good constants can become quite demanding.