Overpartition function modulo 16 and some binary quadratic forms (Q2810669)

An overpartition of \(n\) is an ordered sequence of non-increasing positive integers that sum to \(n\), where the first occurrence of each distinct part may be overlined. The number of overpartitions of \(n\) is denoted by \(\bar{p}(n)\). Various authors have studied congruences for \(\bar{p}(n)\). \textit{B. Kim} [Discrete Math. 309, No. 8, 2528--2532 (2009; Zbl 1228.05046)] gave a complete determination of the overpartition function modulo 8, showing in particular that NEWLINENEWLINE\[ \bar{p}(n) \equiv \begin{cases} 2 \pmod{2^3} & \text{ if }n\text{ is a square of an odd number}, \\ 4 \pmod{2^3} & \text{ if }n\text{ is a double of a square},\\ 6 \pmod{2^3} & \text{ if }n\text{ is a square of an even number},\\ 0 \pmod{2^3} & \text{ otherwise}.\end{cases} \]NEWLINENEWLINESeveral other authors have also studied congruences for \(\bar{p}(n)\) modulo small powers of 2. \textit{O. X. M. Yao} and \textit{E. X. W. Xia} [J. Number Theory 133, No. 6, 1932--1949 (2013; Zbl 1275.11136)] and \textit{W. Y. C. Chen} et al. [Ramanujan J. 40, No. 2, 311--322 (2016; Zbl 1355.11101)] have established a variety congruences for \(\bar{p}(n)\) modulo 16.NEWLINENEWLINE\textit{K. Mahlburg} [Discrete Math. 286, No. 3, 263--267 (2004; Zbl 1114.11083)] and \textit{B. Kim} [Integers 8, No. 1, Article A38, 8 p. (2008; Zbl 1210.11108)] have established density results for \(\bar{p}(n)\) modulo certain powers of 2.NEWLINENEWLINEIn the present work, the author gives a complete determination of \(\bar{p}(n)\) modulo 16, which extends Kim's result modulo 8. The author is able not only to give new proofs of the congruences of Yao and Xia and Chen-Hou-Sun-Zhang, but also uses the methods of the paper to provide infinitely many new Ramanujan-type congruences for overpartitions modulo 16. X.~Xiong's method relates \(\bar{p}(n)\) to certain binary quadratic forms. Extensive use is made of \(e_{2}(n)=\) the number of representations of \(n\) as the form \(m^{2} + 2l^{2}\) with \(m\geq1,\) \(l\geq1\) and \(r_{2}{\prime}(n)=\) the number of representations of \(n\) as a sum of two squares \(m^{2} + l^{2}\) but with \(m \geq \), \(l \geq 1,\) and \(m \neq l\) along with the overpartition function \(\bar{p}(q) = \sum_{n \geq 0} \bar{p}(n) q^{n}\).