Forgotten conjectures of Andrews for Nahm-type sums

Abstract: In his famous '86 paper, Andrews made several conjectures on the function $\sigma(q)$ 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 $\sigma$ has been related to the arithmetic of $\mathbb{Z}[\sqrt{6}]$ 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 $\sigma$, but which also exhibit extraordinary behaviour. In an ongoing project with Folsom, Rolen, and Storzer, we study the function $v_1(q)$ 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 $v_1$, which require a blend of novel and well-known techniques, and reveal that $v_1$ should be intimately linked to the arithmetic of the imaginary quadratic field $\mathbb{Q}[\sqrt{-3}]$.

combinatoricsnumber theory

Audience: researchers in the topic

Lethbridge number theory and combinatorics seminar