On the infinite Borwein product raised to a real power

Abstract: We study the $q$-series coefficients appearing in the expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $\delta$. Application of the Hardy-Ramanujan-Rademacher circle method gives an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture we made concerning an observed sign pattern of the coefficients when the exponent $\delta$ is within a specified range of positive real numbers. We then take a closer look at the cube of the infinite Borwein product, for arbitrary $p$ (now a positive integer), and establish some vanishing and divisibility properties of the respective coefficients. This is joint work with Nian Hong Zhou.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar