Forgotten conjectures of Andrews for Nahm-type sums

Speaker: Joshua Males, University of Manitoba.

Date: Mon, Mar 20, 2023

Location: Online, PIMS, University of Lethbridge

Conference: Lethbridge Number Theory and Combinatorics Seminar

Subject: Mathematics

Class: Scientific


In his famous '86 paper, Andrews made several conjectures on the function σ(q) of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by Andrews-Dyson-Hickerson in their '88 Invent. paper, and the function σ has been related to the arithmetic of Z[6–√]by Cohen (and extensions by Zwegers), and is an important first example of quantum modular forms introduced by Zagier.

A closer inspection of Andrews' '86 paper reveals several more functions that have been a little left in the shadow of their sibling σ , but which also exhibit extraordinary behaviour. In an ongoing project with Folsom, Rolen, and Storzer, we study the function v1(q) which is given by a Nahm-type sum and whose coefficients count certain differences of partition-theoretic objects. We give explanations of four conjectures made by Andrews on v1, which require a blend of novel and well-known techniques, and reveal that v1 should be intimately linked to the arithmetic of the imaginary quadratic field Q[−3−−−√]

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