Certain eta-quotients and arithmetic density of Andrews' singular overpartitions
Singular overpartitions were introduced by \textit{G. E. Andrews} [Int. J. Number Theory 11, No. 5, 1523--1533 (2015; Zbl 1325.11107)] in his study of analogues of the Rogers-Ramanujan identities. In particular, Andrews' singular overpartition function \(\overline{C}_{k,i}(n)\) counts the number of overpartitions of \(n\) in which no part is divisible by \(k\) and only parts congruent to \(\pm i\) modulo \(k\) may be overlined. The object of this paper under review is to investigate the arithmetic densities of \(\overline{C}_{3\ell,\ell}(n)\) modulo arbitrary powers of \(2\) and \(3\) when \(\ell=p\cdot 2^\alpha\) with \(p\) prime, by applying the modularity of certain eta-quotients. For example, the authors show that, for all \(\alpha\ge 1\) with \(2^\alpha\ge p\), the set \(\{n\in\mathbb{N}:\overline{C}_{3p\cdot 2^\alpha,p\cdot 2^\alpha}(n) \equiv 0 \pmod{2^k}\}\) has arithmetic density \(1\).
scientific article; zbMATH DE number 7391387