Where do the maximum absolute $q$-series coefficients of $(1-q)(1-q^2)(1-q^3)\dots(1-q^{n-1})(1-q^n)$ occur?

TL;DR

This paper investigates the location of maximum absolute coefficients in a specific q-series polynomial, using extensive computational data to conjecture a periodic pattern related to the polynomial's degree.

Contribution

The authors conducted large-scale computational experiments to identify a periodic pattern in the maximum coefficient locations of these polynomials, proposing a new conjecture based on their findings.

Findings

Identified a periodic pattern with period 62,624 in the maximum coefficient locations.

Extended computational analysis to n ≤ 75,000.

Formulated conjectures on the distribution of maximum coefficients.

Abstract

We used the MACH2 supercomputer to study coefficients in the $q$-series expansion of $(1-q)(1-q^2)\dots(1-q^n)$, for all $n\leq 75000$. As a result, we were able to conjecture some periodic properties associated with the before unknown location of the maximum coefficient of these polynomials with odd $n$. Remarkably the observed period is 62,624.