Odd values of the Klein j-function and the cubic partition function

TL;DR

This paper establishes a new algebraic lower bound on the count of odd values of the Klein j-function and cubic partition function, improving previous results without relying on modular forms, and further refines bounds using modular form techniques.

Contribution

It provides a novel algebraic approach to bound odd values of the Klein j-function, improving prior results, and demonstrates how modular forms can refine these bounds.

Findings

Lower bound for odd Klein j-function values: (rac{\u221a{x}\, \u2202 ext{loglog}x}{ ext{log}x})

Algebraic methods can establish bounds without modular forms

Modular form techniques can further refine these bounds

Abstract

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.