AI 中文总结
本文精确确定了有限绝对值余弦和的最小值，除少数例外值外均为\\(\lfloor n/2\rfloor\\)，并分类了最小点，证明采用初等分段凹性方法。
AI 中文摘要
对于正整数\\(n\\)，设\\(M_n=\min_{x\in\mathbb R}\sum_{k=1}^n |\cos(kx)|\\)。我们精确确定了\\(M_n\\)的值。除例外值\\(M_2=1/\sqrt2\\)、\\(M_4=1+\sqrt3/2\\)和\\(M_6=(-1+3\sqrt5+2\sqrt{5+2\sqrt5})/4\\)外，有\\(M_n=\lfloor n/2\rfloor\\)。最小点也被分类：对于\\(n\notin\{2,4,6\}\\)，等式仅在\\(x\equiv \pi/2\pmod{\pi}\\)处取得；而例外情形分别在\\(x\equiv\pm\pi/4\\)、\\(\pm\pi/6\\)和\\(\pm\pi/10\pmod{\pi}\\)处取得。证明是初等的，使用了分段凹性、模\\(2q\\)的置换以及两个有限三角估计。
英文摘要
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.
Comments9 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
