A note on the equidistribution of $3$-colour partitions
Joshua Males
TL;DR
The note analyzes equidistribution properties of three families of three-color partitions by deriving sharp near $0$ asymptotics for the generating products $F_{a,c}( heta; e^{-z})$ and applying Wright's circle method. By factoring the relevant generating functions and examining their behavior on the major arc, the authors show that the dominant contribution comes from the principal part and that the residue classes become asymptotically equidistributed across $s$. They obtain explicit asymptotic formulas for $J_E(r,s;n)$ (when $3 mid s$), $J_T(r,s;n)$, and $J_G(r,s;n)$, and deduce the classical asymptotics $J_E(n)=p(n)$ and $J_G(n)=ar p(n)$ as corollaries. The results provide both new equidistribution statements and a versatile analytic toolkit for studying similar partition-analytic problems.
Abstract
In this short note, we prove equidistribution results regarding three families of three-colour partitions recently introduced by Schlosser and Zhou. To do so, we prove an asymptotic formula for the infinite product $F_{a,c}(ζ; {\rm e}^{-z}) := \prod_{n \geq 0} \big(1- ζ{\rm e}^{-(a+cn)z}\big)$ ($a,c \in \mathbb{N}$ with $0<a\leq c$ and $ζ$ a root of unity) when $z$ lies in certain sectors in the right half-plane, which may be useful in studying similar problems. As a corollary, we obtain the asymptotic behaviour of the three-colour partition families at hand.