Asymptotic formula for the $M_2$-ranks of overpartitions

Abstract

Let ${\overline{N}}_2(a,c,n)$ be the number of overpartitions of $n$ whose the $M_2$-rank is congruent to $a$ modulo $c$. In this paper, we obtain the asymptotic formula of ${\overline{N}}_2(a,c,n)$ utilizing the Ingham Tauberian Theorem. As applications, we derive inequalities concerning with ${\overline{N}}_2(a,c,n)$ including its strict concavity and log-concavity.