Oscillating asymptotics and conjectures of Andrews

Abstract In 1986, Andrews (Am Math Mon 93(9):708–711, 1986; Adv Math 61(2):156–164, 1986) studied the function $$\sigma (q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\mathbb {Q}(\sqrt{6})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mn>6</mml:mn> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ; 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 $$\sigma (q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> reveals additional related functions and conjectures, which we study in this paper. In particular, we study the function $$v_1(q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , also from Ramanujan’s “Lost” Notebook, a q -hypergeometric series with partition-theoretic Fourier coefficients $$V_1(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , and prove two of Andrews’ conjectures on $$V_1(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$V_1(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> to the arithmetic of $$\mathbb {Q}(\sqrt{-3})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .