Asymptotic formula for the $M_2$-ranks of overpartitions

TL;DR

This paper derives an asymptotic formula for the distribution of the $M_2$-rank in overpartitions and explores related inequalities, advancing understanding of overpartition rank statistics.

Contribution

It introduces an asymptotic formula for overpartition $M_2$-rank counts using the Ingham Tauberian Theorem, a novel application in this context.

Findings

Asymptotic formula for $ar{N}_2(a,c,n)$ derived.

Established inequalities including strict concavity and log-concavity.

Enhanced understanding of overpartition $M_2$-rank distribution.

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.