  In this note, using entirely algebraic or elementary methods, we determine a
new asymptotic lower bound for the number of odd values of one of the most
important modular functions in number theory, the Klein $j$-function. Namely,
we show that the number of integers $n\le x$ such that the Klein $j$-function
--- or equivalently, the cubic partition function --- is odd is at least of the
order of $$\frac{\sqrt{x} \log \log x}{\log x},$$ for $x$ large. This improves
recent results of Berndt-Yee-Zaharescu and Chen-Lin, and approaches
significantly the best lower bound currently known for the ordinary partition
function, obtained using the theory of modular forms. Unlike many works in this
area, our techniques to show the above result, that have in part been inspired
by some recent ideas of P. Monsky on quadratic representations, do not involve
the use of modular forms.
  Then, in the second part of the article, we show how to employ modular forms
in order to slightly refine our bound. In fact, our brief argument, which
combines a recent result of J.-L. Nicolas and J.-P. Serre with a classical
theorem of J.-P. Serre on the asymptotics of the Fourier coefficients of
certain level 1 modular forms, will more generally apply to provide a lower
bound for the number of odd values of any positive power of the generating
function of the partition function.
