Forgotten conjectures of Andrews for Nahm-type sums

Joshua Males (University of Manitoba, Canada)

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−−−√]
.