= Uniform bound for harmonic sine polynomials
{title2=$\sup_m\left\|\sum_{k=1}^m\frac{\sin kt}{k}\right\|_\infty\leq1+\pi$}
For $0<t\leq\pi$, split the sum at $\lfloor1/t\rfloor$. The first part is bounded using $|\sin kt|\leq kt$; <summation by parts> and the geometric bound on interval sine sums control the rest. The harmonic coefficient sum grows like $\log m$, but cancellations keep the whole <trigonometric polynomial> uniformly bounded. Modulating this polynomial lets a <Fourier partial sum> expose the large uncancelled half.
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