无限乘积实幂的符号模式：Schlosser 与 Zhou 四个猜想的解决
Sign patterns of real powers of infinite products: resolution of four conjectures of Schlosser and Zhou
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中文总结 AI 辅助
本文完全解决了Schlosser与Zhou关于无限乘积实幂系数符号模式的四个猜想，精确刻画了三个猜想失败的确切区间，并揭示了圆法主导项消失的失败机制。
中文摘要 AI 辅助
对于无限乘积 P(q) = prod_{m>=1} (1-q^m)^{eps(m)} 和实数 delta，记 P(q)^delta = sum_{n>=0} c_delta(n) q^n。Schlosser 和 Zhou 针对若干乘积及 delta 的取值范围，猜想这些系数的精确符号模式。我们完全解决了其中的四个猜想，在每种情况下都推广了文献中已有的部分结果：涉及 Goellnitz-Gordon 乘积 Q_8 = (q,q^7;q^8)_inf/(q^3,q^5;q^8)_inf（猜想21）、Q_12 = (q,q^11;q^12)_inf/(q^5,q^7;q^12)_inf（猜想24），以及 Borwein 乘积 G_7 和 G_11（猜想20和23）。猜想23成立。猜想21恰好在区间 [beta, 8/3) 上失败，其中 beta 约为 2.66448；阈值 8/3 是精确的，首批反例出现在 n 约为 7.6 x 10^5 处。猜想24恰好在区间 (delta_1, 0) 上失败，其中 delta_1 = -0.64411... 是第22个系数多项式的根。猜想20恰好在区间 (delta_c, 5) 上失败，其中 delta_c = 4.8735075853867342634... 是第897个系数多项式的根。所有其他声明的取值范围均被证明。猜想21的部分内容由 He 和 Li 独立解决；本文的新贡献在于 delta 接近 -1 的范围，在该处他们的阈值发散。每个失败案例都有相同的机制：一个主导的圆法振幅消失，而一个符号错误的次级项超过它。证明除经典事实外完全自足。尖点分析通过 Weil 表示的有限轨道精确完成；Hardy-Ramanujan-Rademacher 展开被完全显式化；精确算术和区间算术中的认证计算处理了有限范围和退化情形。
英文摘要
For an infinite product P(q) = prod_{m>=1} (1-q^m)^{eps(m)} and real delta, write P(q)^delta = sum_{n>=0} c_delta(n) q^n. Schlosser and Zhou conjectured precise sign patterns for these coefficients, for several products and ranges of delta. We resolve four of their conjectures completely, in each case extending partial results already in the literature: those for the Goellnitz-Gordon product Q_8 = (q,q^7;q^8)_inf/(q^3,q^5;q^8)_inf (Conjecture 21), for Q_12 = (q,q^11;q^12)_inf/(q^5,q^7;q^12)_inf (Conjecture 24), and for the Borwein products G_7 and G_11 (Conjectures 20 and 23). Conjecture 23 is true. Conjecture 21 fails exactly on [beta, 8/3), where beta is approximately 2.66448; the threshold 8/3 is sharp, and the first counterexamples occur near n = 7.6 x 10^5. Conjecture 24 fails exactly on (delta_1, 0), where delta_1 = -0.64411... is a root of the 22nd coefficient polynomial. Conjecture 20 fails exactly on (delta_c, 5), where delta_c = 4.8735075853867342634... is a root of the 897th coefficient polynomial. All other stated ranges are proved. Parts of Conjecture 21 were settled independently by He and Li; what is new here is the range near delta = -1, where their threshold diverges. Each failure has the same mechanism: a leading circle-method amplitude vanishes, and a secondary term with the wrong sign overtakes it. The proofs are self-contained apart from classical facts. The cusp analysis is exact, via finite orbits of Weil representations; the Hardy-Ramanujan-Rademacher expansion is made fully explicit; and certified computations in exact and ball arithmetic handle the finite ranges and the degenerate regimes.
发表机构
- Purdue University（普渡大学）
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