Asymptotics of Nahm sums at roots of unity

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.