Positivity of eta products -- a certain case of K. Saito's conjecture (Q2580242)

Let \(\eta(\tau)\) be the Dedekind eta function. K. Saito conjectured that for any natural number \(h\) the function \[ \frac{\eta(h\tau)^{\varphi(h)}} {\prod_{d| h}\eta(d\tau)^{\mu(d)}}, \] has only non-negative Fourier coefficients, where \(\varphi(h)\) is the Euler function and \(\mu(d)\) is the Möbius function. The author proves this when \(h\) is a power of a prime number.