Exact Minima of Finite Absolute Cosine Sums

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 \(x\equiv π/2\pmodπ\), while the exceptional cases are attained at \(x\equiv\pmπ/4\), \(\pmπ/6\), and \(\pmπ/10\pmodπ\), respectively. The proof is elementary and uses piecewise concavity, a permutation modulo \(2q\), and two finite trigonometric estimates.

arXiv comment9 pages. Revised the introduction and expanded the references to clarify prior results and the attribution of the proof reductions. Added explicit estimates for the small-parameter cases and further details of the minimizer classification. The main theorem and all stated minimum values remain unchanged