Asymptotic formula for the $M_2$-ranks of overpartitions

Abstract

Let 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 N 2 (a, c, n) utilizing the Ingham Tauberian Theorem. As applications, we derive inequalities concerning with N 2 (a, c, n) including its strict concavity and log-concavity.