Note on a problem of Ramanujan

Abstract

For fixed positive real numbers $\omega , \omega '$, it is known that the number of lattice points $(u,v), u\ge 0, v \ge 0$ satisfying $0 \le u \omega + v\omega ' \le \eta$ is given by $\frac{1}{2}\big (\frac{\eta ^{2}}{\omega \omega ^{'}}+\frac{\eta }{\omega } +\frac{\eta }{\omega ^{'}}\big )+ O_{\varepsilon }(\eta ^{1-\frac{1}{\alpha _{0}} +\varepsilon })$, where $\alpha _0 \ge 1$ is a constant. In this paper, we explicitly compute $\alpha _0$ for certain values of $\omega /\omega '$. In particular, in Ramanujan’s case (i.e., when $\omega = \log 2$ and $\omega ' = \log 3$), we show that $\alpha _0 = 2^{18}\log 3$ is admissible. This improves an earlier result of the paper (Ramachandra K, Sankaranarayanan A and Srinivas K, Hardy Ramanujan J. 19 (1996) 2–56), where it was shown that $\alpha _0 = 2^{40}\log 3$ holds.