Asymptotic formula for the $M_2$-ranks of overpartitions

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.

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