  There is not much that can be said for all $x$ and for all $n$ about the sum
\[ \sum_{k=1}^n \frac{1}{|\sin k\pi x|}. \] However, for this and similar sums,
series, and products, we can establish results for almost all $x$ using the
tools of continued fractions. We present in detail the appearance of these sums
in the singular series for the circle method. One particular interest of the
paper is the detailed proof of a striking result of Hardy and Littlewood, whose
compact proof, which delicately uses analytic continuation, has not been
written freshly anywhere since its original publication. This story includes
various parts of late 19th century and early 20th century mathematics.
