  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.
