Asymptotic Formulas Related to the M₂-rank of Partitions without Repeated Odd Parts

2018/05/29 by Jennings-Shaffer, Chris, Reihill, Dillon
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.11319

Abstract

We give asymptotic expansions for the moments of the M₂-rank generating function and for the M₂-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 M₂-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 M₂-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.

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