New density results and congruences for Andrews' singular overpartitions
\textit{G. E. Andrews} [Int. J. Number Theory 11, No. 5, 1523--1533 (2015; Zbl 1325.11107)] introduced the singular overpartitions in order to give overpartition analogues of Rogers-Ramanujan-type theorems for the ordinary partition function with restricted successive ranks. The Andrews singular overpartition function \(\overline{C}_{k,i}(n)\) denotes the number of overpartitions of \(n\) in which no part is divisible by \(k\) and only parts \(\equiv\pm i\pmod{k}\) could be overlined. The generating function of \(\overline{C}_{k,i}(n)\) is given by \[ \sum_{n=0}^\infty \overline{C}_{k,i}(n)q^n=\dfrac{(q^k;q^k)_\infty(-q^i;q^k)_\infty(-q^{k-i};q^k)_\infty}{(q;q)_\infty}, \] where \[ (a;q)_\infty=\prod_{k=0}^\infty(1-aq^k),\qquad|q|<1. \] Soon after, \textit{S. Chen} et al. [Int. J. Number Theory 11, No. 5, 1463--1476 (2015; Zbl 1320.05010)] obtained a complete parity characterization of \(\overline{C}_{4,1}(n)\) and \(\overline{C}_{6,2}(n)\). In this paper, the authors further investigate parity properties of \(\overline{C}_{4k,k}(n)\) and \(\overline{C}_{6k,k}(n)\). More precisely, they prove that for any fixed integer \(j>0\), \[ \begin{multlined}\lim_{X\rightarrow\infty}\dfrac{\#\{0\leq n\leq X:~\overline{C}_{4k,k}(n)\equiv0\pmod{2^j}\}}{X}\\ =\lim_{X\rightarrow\infty}\dfrac{\#\{0\leq n\leq X:~\overline{C}_{6l,l}(n)\equiv0\pmod{3^j}\}}{X}=1,\end{multlined} \] where \(k=2^\alpha m\)~(\(\alpha\geq0\), \(\gcd(m,2)=1\)), and \(2^\alpha>m\); \(l=3^\alpha m\)~(\(\alpha\geq0\), \(\gcd(m,3)=1\)), and \(3^\alpha>4m\). On the other hand, utilizing a result of \textit{K. Ono} and \textit{Y.~Taguchi} [Int. J. Number Theory, No. 1, 75--101 (2005; Zbl 1084.11014)] on nilpotency of Hecke operators, the authors derive two infinite families of congruences modulo powers of 2 enjoyed by \(\overline{C}_{4\cdot2^\alpha,2^\alpha}(n)\) \((\alpha\geq0)\) and \(\overline{C}_{4\cdot3\cdot2^\alpha,3\cdot2^\alpha}(n)\) \((\alpha\geq2)\). More precisely, they prove that there is an integer \(c\geq0\) such that for every \(d\geq1\) and distinct primes \(p_1,\ldots,p_{c+d}\) coprime to \(6\), \[ \overline{C}_{4\cdot2^\alpha,2^\alpha}{\left(\dfrac{p_1\cdots p_{c+d}\cdot n+1-3\cdot2^\alpha}{24}\right)}\equiv0\pmod{2^{d}}, \] where \(\gcd(n,p_1\cdots p_{c+d})=1\). They also obtain a similar result for the function \(\overline{C}_{4\cdot3\cdot2^\alpha,3\cdot2^\alpha}(n)\).
scientific article; zbMATH DE number 7391378