有限傅里叶对偶、径向值与一个模八假 theta 商式的算术
Finite Fourier Duality, Radial Values, and Arithmetic of a Mod Eight False Theta Quotient
- Pie Mathematics Association（Pie数学协会）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究模八假 theta 矩商式的径向行为，建立有限傅里叶闭包，证明周期狄利克雷值非零，并给出系数整除性、二进插值及欧拉变换正性等算术结果。
AI中文摘要：
我们研究一族模八假 theta 矩商式，并确定它们在每个单位根处的径向行为。主要结构结果建立了相关周期边界序列的精确有限傅里叶闭包。该闭包导出一个完整的函数方程，并证明了相关周期狄利克雷值在所有负奇整数处非零。因此，每个商式要么表现出普遍的规范化阶乘渐近，要么具有有限的非零径向值。这些有限值属于分圆域，满足伽罗瓦协变性，并在主根处特化为带符号的奇 Springer 数。我们还确定了精确的系数整除性，构造了一个最优的逐系数二进插值，并识别出公分母的唯一主导零点。该零点给出一个无条件的系数分解。最后，利用精确边界估计和莫比乌斯反演，我们证明了分母的正式欧拉变换中每个指数均为正。
英文摘要：
We study a family of mod-eight false-theta moment quotients and determine their radial behaviour at every root of unity. The main structural result establishes an exact finite Fourier closure for the associated periodic boundary sequences. This closure yields a completed functional equation and proves the nonvanishing of the relevant periodic Dirichlet values at all negative odd integers. Consequently, each quotient exhibits either universal normalized factorial asymptotics or a finite nonzero radial value. The finite values belong to cyclotomic fields, satisfy Galois covariance, and specialize at the principal root to signed odd Springer numbers. We also determine exact coefficient divisibility, construct an optimal coefficientwise two-adic interpolation, and identify the unique dominant zero of the common denominator. This zero gives an unconditional coefficient decomposition. Finally, using exact boundary estimates and Möbius inversion, we prove positivity of every exponent in the formal Euler transform of the denominator.
