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Asymptotics Of Nahm Sums And Conjectures From Ramanujan's 'Lost' Notebook.pdf

2305.16654.pdf
Preview of Asymptotics of Nahm Sums and Conjectures from Ramanujan's 'Lost' Notebook
🔗 Source: arxiv.org
📊 Size: 726 KB
📄 Pages: 35 pages
⬇️ Downloads: 31

Summary

Both functions are q-hypergeometric series with combinatorial interpretations involving partitions.

Key Points:

1. Andrews' Conjectures: In 1986, Andrew made several conjectures about the Fourier coefficients S(n) of σ(q) and V1(n) of v1(q), predicting their growth behavior and sign patterns.

2. Connection to Q(√6): Andrews-Dyson-Hickerson (1988) resolved these conjectures by relating the coefficients to the arithmetic of Q(√6). This was further expanded upon by Cohen (1992) using Maass waveforms, and more recently by Zwegers and Li-Roehrig.

3. New Results: The authors prove two of Andrews' conjectures for V1(n), showing that |V1(n)| goes to infinity away from a set of density 0 (Conjecture 3) and that the signs of V1(n), V1(n+1), V1(n+2), and V1(n+3) are mostly regular (Conjecture 4).

4. Asymptotic Behavior: They establish the asymptotic behavior of v1(q), which leads to understanding the sign patterns in V1(n). The sequence V1(n) exhibits recurring sections with specific sign patterns depending on n modulo 4.

5. Additional Conjectures: The paper also discusses additional related conjectures of Andrews, connecting the asymptotics of V1(n) to the arithmetic of Q(√-3).

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Description

It builds upon previous resolutions linking Fourier coefficients to arithmetic of Q(√6) and extends these connections to a new function v1(q) with its own partition-theoretic coefficients V1(n).

Technical Information

  • File Format: PDF
  • File Size: 726 KB
  • Pages: 35
  • Language: EN
  • Total Downloads: 31
  • Last Updated: 4 days ago

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