Odd values of the Rogers-Ramanujan functions
In this paper, the author considers \[ \gamma(N)= \# \{1\leq n \leq N: n\equiv 1\pmod 2\text{ and } g(n) \equiv 1 \pmod 2 \} \] and \[ \delta(N)= \# \{1\leq n \leq N: n\equiv 0\pmod 2\text{ and } h(n) \equiv 1 \pmod 2 \}, \] where \(g(n)\), respectively \(h(n)\) denote the the number of partitions of the positive integer \(n\) into parts \(\equiv \pm1 \bmod 5\), respectively \(\pm2 \bmod 5\). \textit{B. Gordon} [Dev. Math. 23, 83--93 (2012; Zbl 1264.11090)] proved the order of magnitude of \(\gamma(N)\) and \(\delta(N)\) is \(\frac{N}{\log N}\). For sufficiently large \(N\), the author proves that \[ \gamma(N)=\frac{\pi^2}{5}\cdot \frac{N}{\log N} + O\left(\frac{N \log \log N}{\log^2 N} \right) \] and \[ \delta(N)=\frac{\pi^2}{5}\cdot \frac{N}{\log N} + O\left(\frac{N \log \log N}{\log^2 N} \right), \] which improve Gordon's result.
scientific article; zbMATH DE number 6994049