Asymptotics of Nahm sums at roots of unity Article Swipe

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Stavros Garoufalidis , Don Zagier ·

YOU? · · 2020 · Open Access · · DOI: https://doi.org/10.1007/s11139-020-00266-x · OA: W3044926176

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.