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

Abstract

Klein’s modular j-function is defined to be j(z) = E 4/Δ(z) = 1 q + 744 + ∞ ∑ n=1 c(n)q where z ∈ C with (z) > 0, q = exp(2iπz), E4(z) denotes the normalized Eisenstein series of weight 4 and Δ(z) is the Ramanujan’s Delta function. In this short note, we show that for each integer a ≥ 1, the interval (a, 4a(a+1)) (respectively, the interval (16a−1, (4a+1)2)) contains an integer n with n ≡ 7 (mod 8) such that c(n) is odd (respectively, c(n) is even).