Is $\sum_{n=1}^N e^{2 \pi p_n z i}$ bounded for irrational $z$?

Question

Let $p_n$ be the $n$th prime number. If $z$ is irrational real, is it known whether the partial sums $\sum_{n=1}^N e^{2 \pi p_n z i}$ are bounded? It seems the partial sums are unbounded if $z$ is rational.

Answer 1

This is essentially the exponential sum used by Vinogradov in his proof of the odd Goldbach conjecture (the same sum has been used many times since). There is an exact formula for the value of the sum at rational numbers, and a good approximation for irrational numbers that are quite close to rational numbers (these are points on the "major arcs"). So if you took an irrational number $z$ with an infinite sequence of very good rational approximations, you could probably get enough information on the size of the sum to conclude that it is unbounded. (Presumably it is infinitely often as large as a constant times $\sqrt N$. This is probably true even on the "minor arcs", but I don't know whether it's easy to prove.)