Arithmetic properties of Schur-type overpartitions
In the theory of partitions there are numerous results like Schur's theorem and the Rogers-Ramanujan identities. It is natural to ask about the parity or other arithmetic properties of the counting functions associated with these identities. In this paper, the authors consider a certain counting function \(S(n)\), which first arose in an overpartition identity related to Schur's partition theorem and a universal mock theta function in [\textit{K. Bringmann} et al., Math. Res. Lett. 23, No. 1, 67--80 (2016; Zbl 1342.05010)]. Using basic facts about modular forms and some elementary algebraic number theory, the authors provide a complete characterization of the parity of this overpartition function in the progressions \(2n+1\), \(4n+2\), and \(8n+4\), a complete characterization of the residue modulo \(4\) in the progressions \(8n + 5\) and \(8n + 7\), and some Ramanujan-type congruences modulo \(5\).
scientific article; zbMATH DE number 7635042