伽罗瓦轨道结构、拉马努金和与分圆余弦公式的稳定范围坍缩
Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas
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中文总结 AI 辅助
本文结合伽罗瓦轨道与有限频率展开，研究分圆余弦求值，揭示值层面伽罗瓦不变性缺乏稳定范围刚性，并证明特定模式族在足够大水平下坍缩为仿射函数，最终用拉马努金和给出精确轨道公式。
中文摘要 AI 辅助
我们通过将伽罗瓦轨道结构与有限频率展开相结合，研究超越完全对称设置的分圆余弦求值。首先，对于定义在有理数域上的截断兼容多项式族，我们区分了求值不变性与多项式公式不变性。由水平的某个有理函数可求和性迫使求值最终为有理数。反之，每个有理值序列都可以由一个兼容族实现，其总次数一致地以二为界，这表明仅凭值层面的伽罗瓦不变性几乎不携带稳定范围的刚性。随后，我们引入由乘法配置索引的显式轨道结构模式求和。对于每个固定模式，在每个足够大的允许水平n处，其分圆余弦求值坍缩为仿射函数：n乘以S0(r,h)减去2的|h|次幂，其中S0是相关洛朗多项式的零频率系数。因此，由有限个此类模式通过单一多项式配方统一生成的每个族最终具有多项式求值。最后，将穿孔索引集分解为最大公约数轨道，得到用拉马努金和表示的精确轨道公式。半素数和素数幂情形成为该一般恒等式的透明特例。这些结果既识别了从值层面伽罗瓦不变性扩展对称刚性的障碍，也提供了一个表现出均匀稳定范围坍缩的具体非对称类。
英文摘要
We study cyclotomic cosine evaluations beyond the fully symmetric setting by combining Galois-orbit structure with finite-frequency expansions. First, for truncation-compatible poly?nomial families over Q, we distinguish invariance of the evaluated value from invariance of the polynomial formula. Summability by a rational function of the level forces eventual rationality of the evaluated values. Conversely, every rational-valued sequence can be realized by a compatible family whose total degree is uniformly bounded by two, showing that value-level Galois invariance alone carries essentially no stable-range rigidity. We then introduce explicit orbit-structured pattern sums indexed by multiplicative configurations. For every fixed pattern, at each sufficiently large admissible level n, its cyclotomic cosine evaluation collapses to an affine function n times S0(r,h) minus 2 to the abs(h) power, where S0 is the zero-frequency coefficient of the associated Laurent polynomial. Consequently, every family uniformly generated from finitely many such patterns by a single polynomial recipe has eventually polynomial evaluation. Finally, decomposing the punctured index set into gcd-orbits yields an exact orbitwise formula in terms of Ramanujan sums. The semiprime and prime-power cases become transparent specializations of this general identity. These results identify both the obstruction to extending symmetric rigidity from value-level Galois invariance and a concrete non-symmetric class exhibiting uniform stable-range collapse
发表机构
- Departamento de Matemáticas, Sede Medellín, Universidad Nacional de Colombia（哥伦比亚国立大学麦德林校区数学系）
- Departamento de Matemáticas, Universidad EAFIT（伊法蒂大学数学系）
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