Exact Minima of Finite Absolute Cosine Sums

RESUMO

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.