AI 中文总结
本文证明了Andrews在Ramanujan遗失笔记本中观察到的某些q-级数的符号现象，通过改进的圆法给出精确渐近公式，并推广到无穷族q-超几何级数。
AI 中文摘要
本文证明了Andrews首次在Ramanujan遗失笔记本中某些$q$-级数中观察到的符号现象。对于Andrews考虑的三个级数，即$v_2(q)$、$v_3(q)$和$v_4(q)$，我们证明其系数交替符号，仅存在密度为零的例外集。我们的方法通过改进的圆法给出系数的精确渐近公式，受Folsom-Males-Rolen-Storzer关于$q$-级数$v_1(q)$工作的启发，揭示了指数增长与振荡行为之间的相互作用。这种相互作用产生了一个主导的交替符号因子，支配着Andrews数值观察到的符号规律性。更广泛地，我们为包含这些例子的显式无穷族$q$-超几何级数建立了相同的符号行为，并表明它系统性地源于这些$q$-级数在单位根附近的振荡渐近性。我们引入了一个额外的族，其系数似乎表现出类似的符号规律性，表明这种现象是普遍的，可能指向更深层的理论基础。
英文摘要
In this paper, we prove a sign phenomenon first observed by Andrews for certain $q$-series from Ramanujan's Lost Notebook. For three of the series considered by Andrews, namely $v_2(q)$, $v_3(q)$, and $v_4(q)$, we show that the coefficients are alternating in sign, with only a density-zero set of exceptions. Our approach yields precise asymptotic formulas for the coefficients via an adapted circle method, inspired by the work of Folsom-Males-Rolen-Storzer on the $q$-series $v_1(q)$, revealing an interplay between exponential growth and oscillatory behaviour. This interaction produces a dominant alternating sign factor, which governs the sign regularity observed numerically by Andrews. More broadly, we establish the same sign behaviour for explicit infinite families of $q$-hypergeometric series encompassing these examples, and show that it arises systematically from oscillatory asymptotics of these $q$-series near roots of unity. We introduce an additional family whose coefficients appear to exhibit similar sign regularity, suggesting that this phenomenon is widespread and may point towards a deeper underlying theory.
Commentsv2: minor corrections
