We give a formula for the radial asymptotics to all orders of the special q-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in Calegari et al. (Bloch groups, algebraic K-theory, units and Nahm’s conjecture. arXiv:1712.04887, 2017) to prove Nahm’s conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in K-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann–Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm sums that will be discussed further in a subsequent publication.

我们给出一个特殊的q超几何级数的所有阶的径向渐近公式，这些阶被称为纳姆和，位于单位复数的根上。该结果用于Calegari等人。（Bloch群，代数K-理论，单位和Nahm的猜想。arXiv：1712.04887，2017年）证明Nahm的猜想将Nahm和的模块化与K理论中某个不变式的消失联系起来。一旦将Neumann–Zagier数据转换成Nahm数据，我们渐近公式中出现的幂级数就等于一个结的Kashaev不变量的猜想渐近性，这表明量子结不变量的渐近性与Nahm和的渐近性之间的深层联系在后续出版物中进一步讨论。