This seems not quite in the spirit of the original question, but I couldn't resist mentioning some work which was presented at a recent PIMS (applied) maths colloquium here:
On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations. Kenneth D. T.-R. McLaughlin, Nigel J. E. Pitt. (arXiv 1107.1571)
My limited understanding of the story, which should be taken with a big lump of salt, is as follows:
For certain nonlinear PDEs with a periodic boundary condition, one can write down a Fourier series that represents a weak solution, and then faces the issue of determining in what sense this series converges to the solution. Bizarrely (to my eyes) the shape of the solution is different for rational and irrational times; and to understand what happens at irrational times, the authors have to deal with exponential sums of a form encountered in analytic number theory.
Furthermore, Theorem 1.5 in this paper, which demonstrates a kind of "Gibbs phenomenon" for these solutions, is proved for irrational values of $t$ satisfying a complicated condition depending on the continued fraction expansion of $t$.
(According to the speaker (McLaughlin), the collaboration started while he was sitting in number theory lectures given by the first author, reading a PDEs paper, and realizing that the sums on the page in front of him were awfully like the sums on the board.)