Oscillating asymptotics and conjectures of Andrews

Abstract

In 1986, Andrews studied the function $\sigma(q)$ from Ramanujan's ``Lost" Notebook, and made several conjectures on its Fourier coefficients $S(n)$, which count certain partition ranks. In 1988, Andrews-Dyson-Hickerson famously resolved these conjectures, relating the coefficients $S(n)$ to the arithmetic of $\mathbb Q(\sqrt{6})$; this relationship was further expounded upon by Cohen in his work on Maass waveforms, and was more recently extended by Zwegers and by Li and Roehrig. A closer inspection of Andrews' original work on $\sigma(q)$ reveals additional related functions and conjectures, which we study in this paper. In particular, we study the function $v_1(q)$, also from Ramanujan's ``Lost" Notebook, a $q$-hypergeometric series with partition-theoretic Fourier coefficients $V_1(n)$, and prove two of Andrews' conjectures on $V_1(n)$ which are parallel to his original conjectures on $S(n)$. Our methods differ from those used by Andrews-Dyson-Hickerson, and require a blend of novel techniques inspired by Garoufalidis' and Zagier's recent work on asymptotics of Nahm sums, 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 $V_1(n)$ to the arithmetic of $\mathbb Q(\sqrt{-3})$.