发表机构
Vanderbilt University（范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究拉马努金遗失笔记本中$v_2(q)$、$v_3(q)$和$v_4(q)$这三个$q$级数的特殊符号对现象，结合渐近展开与振荡因子分析，证明特殊指标以结构化方式无限次出现，存在无穷多对同号连续系数且至少一个是局部最小值。
AI 中文摘要
本文证明了拉马努金遗失笔记本中三个$q$级数，即$v_2(q)$、$v_3(q)$和$v_4(q)$的一个特殊符号对现象。这一现象最早由安德鲁斯于1986年的一篇论文中观察到。基于昆杜、斯托泽、王以及作者近期的工作，这些$q$级数的系数除了在一个密度为零的集合外是交替变号的，我们研究了交替符号模式失效的特殊指标。我们证明这些特殊指标以一种结构化的方式无限次出现。具体而言，我们表明存在无穷多对连续系数具有相同符号，并且每对中至少有一个系数是系数绝对值序列的局部最小值。我们的证明将系数的精确渐近展开与对展开式中出现的控制特殊符号行为的振荡因子的仔细分析相结合。
英文摘要
In this paper, we prove an \emph{exceptional sign pair phenomenon} for a general family of integer sequences with exponentially growing oscillatory asymptotics. This resolves, in particular, conjectural sign patterns proposed by Andrews in a 1986 paper for three $q$-series from Ramanujan's Lost Notebook, namely $v_2(q), v_3(q),$ and $v_4(q)$. Recent work of Kundu, Storzer, Wang, and the author established that the coefficients of these $q$-series are alternating in sign except in a density-zero set. We prove here that the exceptional indices where the alternating sign pattern fails occur infinitely often in a structured manner. More precisely, we show that there exist infinitely many pairs of consecutive coefficients having the same sign, and that the magnitude of at least one coefficient in each pair is a local minimum of the sequence of absolute values of the coefficients. Together with earlier results, this completes the resolution of Andrews' conjectures and their analogs for all the $q$-series in his original study.
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