On the number of odd values of the Klein j-function and the cubic partition function

Abstract

Using entirely algebraic or elementary methods, partly inspired by recent ideas of P. Monsky on quadratic representations, we determine a new asymptotic lower bound for the number of odd values of a fundamental modular function in number theory, the Klein j-function. Namely, the number of n≤x such that the Klein j-function — or equivalently, the cubic partition function — is odd has order at leastxlog⁡log⁡xlog⁡x, for x→∞. This improves recent results of Berndt, Yee and Zaharescu and Chen and Lin, approaching significantly the best bound currently known for p(n), obtained using modular forms. Then, in the final section, we show how to employ modular forms to slightly refine our bound. In fact, our brief argument, combining a recent result of Nicolas–Serre with a classical theorem of Serre on level 1 modular forms, more generally provides a lower bound for the odd values of any positive power of the generating function of p(n).