Note on a problem of Ramanujan

For fixed positive real numbers omega,omega ', it is known that the number of lattice points (u,v), u >= 0, v >= 0 satisfying 0 <= u omega + v omega ' <= eta is given by 1/2(eta(2)/omega omega '+eta/omega+eta/omega ')+O-epsilon(eta(1-1 alpha 0+epsilon)), where alpha(0) >= 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.

Automated summary

The paper investigates the number of lattice points satisfying certain conditions for fixed real numbers ω and ω', providing explicit calculations for corresponding α0 values, and improving upon previous results by showing that in Ramanujan's case, α0 = 2(18) log 3 is acceptable, contrary to the earlier suggestion of α0 = 2(40) log 3.