Asymptotic Formulas Related to the $M_2$-rank of Partitions without Repeated Odd Parts

TL;DR

This paper derives asymptotic formulas for the moments and values of the $M_2$-rank generating function for partitions without repeated odd parts, using the Hardy-Ramanujan circle method extended to mock modular forms, and resolves some conjectured inequalities.

Contribution

It extends the Hardy-Ramanujan circle method to mock modular forms to obtain asymptotics for the $M_2$-rank and related partition counts, resolving conjectured inequalities.

Findings

Asymptotic expansions for $M_2$-rank moments

Asymptotics for $M_2$-rank at roots of unity

Resolution of conjectured inequalities by Mao

Abstract

We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method extended to mock modular forms. Our formulas for the $M_2$-rank at roots of unity lead to asymptotics for certain combinations of $N2(r,m,n)$ (the number of partitions without repeated odd parts of $n$ with $M_2$-rank congruent to $r$ modulo $m$). This allows us to deduce inequalities among certain combinations of $N2(r,m,n)$. In particular, we resolve a few conjectured inequalities of Mao.