On the parity of the Fourier coefficients of $j$-function

Abstract:

Klein’s modular $j$-function is defined to be \[ j(z) = E_4^3/ \Delta (z) = \frac {1}q + 744 + \sum _{n=1}^\infty c(n)q^n\] where $z\in {\Bbb C}$ with $\Im (z) > 0$, $q = \exp (2i\pi z)$, $E_4(z)$ denotes the normalized Eisenstein series of weight $4$ and $\Delta (z)$ is the Ramanujan’s Delta function. In this short note, we show that for each integer $a\geq 1$, the interval $(a, 4a(a+1))$ (respectively, the interval $(16a-1, (4a+1)^2)$) contains an integer $n$ with $n\equiv 7\pmod {8}$ such that $c(n)$ is odd (respectively, $c(n)$ is even).