TITLE: Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
QUESTION [40 upvotes]: I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?

REPLY [28 votes]: Terry Tao has already given an excellent answer to this, but I want to point out that much more is known about the partial sums of $\sin(\pi k^2 x)$ with $x$ being an irrational (such as $1/\pi$ in the question).  This and related problems were studied extensively in the classical paper of Hardy and Littlewood Some Problems of Diophantine approximation. II from 1914.  Hardy and Littlewood exploit the connection with the transformation formulae for $\theta$-functions, and show a number of $\Omega$ and $O$ results for such partial sums.  In particular, Theorem 2.30 of their paper proves that for any irrational $x$ the series 
$$ 
\sum n^{-\alpha} \cos(n^2 \pi x), \ \text{and} \ \sum n^{-\alpha} \sin(n^2 \pi x) 
$$ 
are not convergent when $0< \alpha \le 1/2$ (and moreover are not summable by any Cesaro means).  By partial summation, this implies that there are arbitrarily large values $N$ with 
$$ 
\Big| \sum_{k\le N} \sin (k^2 \pi x) \Big| \ge \frac{\sqrt{N}}{(\log N)^2},
$$ 
say (otherwise the series in the Hardy-Littlewood result would converge for $\alpha=1/2$), and in fact one can probably get $\gg \sqrt{N}$ from their paper (they do this explicitly for partial sums of $e^{i\pi n^2 x}$).  Quadratic Weyl sums continue to be of interest -- see this very recent paper of Cellarosi and Marklof.