On partitions of \(N\) into summands coprime to \(N\) (Q1255037)

Let \(R(n)\) and \(R'(n)\) denote the number of partitions of \(n\) into summands and distinct summands respectively that are relatively prime to \(n\). \textit{P.Erdős} has shown [Ann. Math., II. Ser. 43, 437-450 (1942; Zbl 0061.07905)] that \[ \log R(n)\thicksim\pi\sqrt{2/3}\varphi^{1/2}(n),\quad\log R'(n)\thicksim\pi\sqrt{2/3}\varphi^{1/2}(n) \] where \(\varphi(n)\) denotes Euler's function. This paper obtains more explicit and precise results by applying the results of \textit{K. F. Roth} and \textit{G. Szekeres} [Quart. J. Math., Oxf. II. Ser. 5, 241-259 (1954; Zbl 0057.03902)] than obtained by \textit{B. Richmond} [J. Number Theory 9, 525-534 (1977; Zbl 0363.10032)].