Oscillating asymptotics and conjectures of Andrews

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dc.contributor.authorFolsom, Amanda
dc.contributor.authorMales, Joshua
dc.contributor.authorRolen, Larry
dc.contributor.authorStorzer, Matthias
dc.contributor.funderThe Shanks Endowment
dc.contributor.funderMax-Planck-Gesellschaft, MPG
dc.contributor.funderMax Planck Institute for Mathematics, Bonn, Germany
dc.contributor.funderPacific Institute for the Mathematical Sciences, PIMS
dc.contributor.funderVanderbilt University, VU
dc.contributor.funderNSF, DMS-1951393-DMS-2200728
dc.contributor.funderSimons Foundation, SF, 853830, LR
dc.contributor.funderNational Security Agency, NSA, H98230-20-1-0022
dc.date.accessioned2026-07-20T13:50:03Z
dc.date.available2026-07-20T13:50:03Z
dc.date.issued2026-06-09
dc.description.abstractIn 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).en
dc.description.sponsorshipThe first author is grateful for support from NSF Grant DMS-2200728, and also thanks the Max Planck Institute for Mathematics, Bonn, Germany, for its hospitality and support during portions of the writing of this paper. The research of the second author conducted for this paper was partially supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute. This work was supported by a grant from the Simons Foundation (853830, LR). The third author is also grateful for support from a 2021–2023 Dean’s Faculty Fellowship from Vanderbilt University. The fourth author has been supported by the Max-Planck-Gesellschaft. The authors also acknowledge the Vanderbilt University “100 Years of Mock Theta Functions: New Directions in Partitions, Modular Forms, and Mock Modular Forms” Conference in May 2022, at which they discussed the topic of this paper and related ideas. This conference was supported by The Shanks Endowment, Vanderbilt University, NSF Grant: “100 Years of Mock Theta Functions
dc.description.sponsorshipNew Directions in Partitions, Modular Forms, and Mock Modular Forms” award number: DMS-1951393, and NSA grant “100 Years of Mock Theta Functions,” award number: H98230-20-1-0022.
dc.description.versionPublished Version
dc.format.extent48
dc.format.mimetypeapplication/pdfen
dc.identifier.articleid83
dc.identifier.authororcidFolsom, Amanda
dc.identifier.authororcidMales, Joshua
dc.identifier.authororcidRolen, Larry
dc.identifier.authororcidStorzer, Matthias
dc.identifier.citationFolsom, A, Males, J, Rolen, L & Storzer, M 2026, 'Oscillating asymptotics and conjectures of Andrews', Mathematische Annalen, vol. 395, no. 4, 83, pp. 1-48. https://doi.org/10.1007/s00208-026-03446-0
dc.identifier.doi10.1007/s00208-026-03446-0
dc.identifier.endpage48
dc.identifier.issn0025-5831
dc.identifier.issued4
dc.identifier.journaltitleMathematische Annalen
dc.identifier.startpage1
dc.identifier.urihttps://hdl.handle.net/10468/19082
dc.identifier.volume395
dc.language.isoen
dc.publisherSpringer New York
dc.relation.urihttps://www.scopus.com/pages/publications/105041296953
dc.rights© 2026, the Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
dc.rights.accessrightsopen access
dc.rights.licensenameAttribution 4.0 International
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.statusPeer reviewed
dc.subjectOscillating asymptotics
dc.subjectConjectures of Andrews
dc.subject[Maths]
dc.titleOscillating asymptotics and conjectures of Andrewsen
dc.typeArticle (peer-reviewed)
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