Oscillating asymptotics and conjectures of Andrews

In 1986, Andrews (Am Math Mon 93(9):708–711, 1986; Adv Math 61(2):156–164, 1986) studied the function σ(q) from Ramanujan’s “Lost” Notebook, and made several conjectures on its Fourier coefficients S(n), which count certain partition ranks. In 1988, Andrews et al. (Invent Math 91(3):391–407, 1988) famously resolved these conjectures, relating the coefficients S(n) to the arithmetic of Q(6); this relationship was further expounded upon by Cohen (Invent Math 91(3):409–422, 1988) in his work on Maass waveforms, and was more recently extended by Zwegers (Q J Math 63(3):753–770, 2012) and by Li and Roehrig (Q J Math 76:367–380, 2025). A closer inspection of Andrews’ original work on σ(q) reveals additional related functions and conjectures, which we study in this paper. In particular, we study the function v1(q), also from Ramanujan’s “Lost” Notebook, a q-hypergeometric series with partition-theoretic Fourier coefficients V1(n), and prove two of Andrews’ conjectures on V1(n) which are parallel to his original conjectures on S(n). Our methods differ from those used in Andrews et al. (Invent Math 91(3):391–407, 1988), and require a blend of novel techniques inspired by Garoufalidis’ and Zagier’s recent work on asymptotics of Nahm sums (Garoufalidis and Zagier in Ramanujan J 55:1, 2021; SIGMA Symmetry Integr Geom Methods Appl 19:Paper No. 082, 2023), with classical techniques including the Circle Method in Analytic Number Theory; our methods may also be applied to determine the asymptotic behavior of similar q-hypergeometric series of interest which are not amenable to classical techniques. We also offer explanations of additional related conjectures of Andrews, ultimately connecting the asymptotics of V1(n) to the arithmetic of Q(-3).