Abstract
We determine the optimal inequality of the form $$\sum _{k=1}^m a_k\sin kx\le 1$$ , in the sense that $$\sum _{k=1}^m a_k$$ is maximal. We also solve exactly the analogous problem for the sawtooth (or signed fractional part) function. Equivalently, we solve exactly an optimization problem about equidistribution on the unit circle.
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