Partitions and indefinite quadratic forms.
The authors have studied the function \[ \sigma(q)=1+\sum^\infty_{n=1}\frac{q^{n(n+1)/2}}{(1+q)(1+q^2)\cdots (1+q^n)}\sum^\infty_{n=0} S(n)q^n \] which appeared in connection with identities from Ramanujan's ``Lost'' Notebook. In an earlier paper Andrews noted that \(S(n)\) is equal to the number of partitions of \(n\) into distinct parts with even rank minus the number with odd rank. The rank of a partition is the largest part minus the number of parts. In this paper they prove a number of results relating \(S(n)\) to the arithmetic of \(\mathbb Q(\sqrt{6})\). For \(m\equiv 1\pmod {24}\) consider the Pell equation \(u^2-6v^2=m\). If \((u,v)\) and \((u',v')\) are two equivalent solutions then \((u+3v)\equiv\pm(u'+3v')\pmod{12}\). Let \(T(m)\) be the excess of the number of inequivalent solutions with \(u+3v\equiv\pm 1\pmod{12}\) over the number with \(u+3v\equiv\pm 5 \pmod{12}\). The main result is the proof of \(S(n)=T(24n+1)\). It is also proved that \(S(n)\) assumes all integer values infinitely often as well as a generating function counterpart of the main result. An explicit form for \(T(m)\) is obtained which is used to prove that \(\limsup | S(n)|=+\infty\) and that \(S(n)=0\) for infinitely many \(n\). A partition-theoretic interpretation of \(T(m)\) is also given. This paper indicates the surprising interaction between the theory of partitions and algebraic number theory. In the follow-up article [ibid. 91, No. 3, 409--422 (1988; Zbl 0642.10013); see the review below] \textit{H. Cohen} has studied these problems as a natural extension of their results.
scientific article; zbMATH DE number 4045826