Can we give an upper bound for the sum over primes $p_{i}$ of $\sin(p_{i} x)$?

Question

Let $x$ be a positive real number. Consider the sum $\sum \sin(p_i x)$ taken over all primes $p_i$ from 2 till $n$. Call this function $f(n,x)$.

Can we give good upper and lower bounds of $f(n,x)$ for fixed $n$ or $x$? Is the best known upper bound of $f(n,x)$ for sufficiently large $n$:

1. approximately equal to $\sqrt{2n+3\sin^2(x)}$ ?

2. equivalent to GRH ?

3. a consequence of GRH ?

Is it useful to consider taking the power series of $sin(x)$ to help compute an upper bound? It seems natural considering we have an approximation for $\sum$ $p_i^k$. (see my other recent question Generalized PNT ?")

But when we do that, we seem to arrive at a too small upper bound. (or at least when I try it). Perhaps it's just a bad idea to try taylor series for something that grows slower than polynomial ... here ? ( or in general ? )

The analogue question for $cos$.

Answer 1

You are looking for bounds on Exponential sums over prime numbers. There are lots of articles on this subject matter, since as you mention, they can be very useful in analytic number theory. (The circle method for example, or more simply when looking at Fourier transform of the indicator function of the primes) Take a look at this paper by Goldston, it looks at results regarding the sum $$\sum_{p\leq x} e(p\alpha)\log p $$ where $e(p\alpha)=e^{2\pi \alpha i}$. The $\log p$ is put in because it is natural to weight the primes this way, it can be removed with partial summation, and since $e^{2\pi \alpha i}=\cos(2\pi \alpha)+i\sin(2\pi \alpha)$ you can recover your sums above from these results.

Hope that helps,