K. Saito's Conjecture for nonnegative eta products and analogous results for other infinite products (Q927724)

The conjecture in the title asserts the nonnegativity of the Fourier coefficients of the product \[ S_N(\tau)= \eta(N_\tau)^{\phi(N)} \prod_{d|N} \eta(d\tau)^{-\mu(d)}, \] where \(\eta(\tau)\) is the classical Dedekind eta function and \(N\) is a positive integer. Saito and others proved the conjecture for various values of \(N\), and this paper proves it for general \(N\). Analogous results are also obtained for other infinite products.