Constancy of an Infinite Cyclotomic Product via Ramanujan Sums

We show that the infinite product defined by $$P(z)=-\prod _{n=1}^{\mathrm{\infty }}({\mathrm{\Phi }}_n(z){)}^{-1\mathrm{/}n},$$ where ${\mathrm{\Phi }}_n(z)$ is the $n$-th cyclotomic polynomial, is constant inside the unit disk. The proof translates a result of Ramanujan on Ramanujan sums, equivalent to the prime number theorem, to the setting of infinite products. We also show that similar identities proved by Ramanujan lead to additional results on infinite cyclotomic products.