Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$? I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
 A: No.  If one selects a number $k$ at random from $1$ to a large number $n$, then for any fixed $h$, the random variables $\sin((k+1)^2), \dots, \sin((k+h)^2)$ asymptotically have mean zero, variance 1/2, and covariances 0, from standard Weyl sum estimates.  Hence the variance of $\sum_{i=1}^h \sin((k+i)^2)$ is asymptotically $h/2$, which goes to infinity as $h \to \infty$.  On the other hand, if the partial sums of $\sin(k^2)$ were bounded, then this variance would have to be bounded also.  [Exercise: what part of the above argument breaks down when working with $\sin(k)$ instead of $\sin(k^2)$?]
It may be possible to push this argument to show that the partial sums have to fluctuate by $\gg \sqrt{n}$ infinitely often, but I haven't checked this (certainly a lower bound of $\gg n^\varepsilon$ for some small $\varepsilon>0$ should be possible from the above argument, perhaps contingent on some conjecture about the irrationality measure of $\pi$).  Heuristically, the law of the iterated logarithm suggests that the sum can occasionally get as large as $\gg \sqrt{n \log\log n}$, but no larger.
A: 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.
