Exact Minima of Finite Absolute Cosine Sums

Ruixi Sun

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arXiv · 2609.05545

Exact Minima of Finite Absolute Cosine Sums

Ruixi Sun

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Abstract

For a positive integer $n$, let $M_n=\min_{x\in\mathbb R}\sum_{k=1}^n |\cos(kx)|$. We determine $M_n$ exactly. Apart from the exceptional values $M_2=1/\sqrt2$, $M_4=1+\sqrt3/2$, and $M_6=(-1+3\sqrt5+2\sqrt{5+2\sqrt5})/4$, one has $M_n=\lfloor n/2\rfloor$. The minimizers are also classified: for $n\notin\{2,4,6\}$, equality is attained only at $x\equiv \pi/2\pmod{\pi}$, while the exceptional cases are attained at $x\equiv\pm\pi/4$, $\pm\pi/6$, and $\pm\pi/10\pmod{\pi}$, respectively. The proof is elementary and uses piecewise concavity, a permutation modulo $2q$, and two finite trigonometric estimates.

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Ruixi Sun. 2026-09-02. Exact Minima of Finite Absolute Cosine Sums. https://arxiv.org/abs/2609.05545

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