Constancy of an Infinite Cyclotomic Product via Ramanujan Sums
Hartosh Singh Bal
TL;DR
This paper proves that the infinite cyclotomic product $P(z) = -\prod_{n=1}^{\infty} (\Phi_n(z))^{-1/n}$ is constant inside the unit disk, with $P(z)\equiv 1$ for $|z|<1$. The proof translates Ramanujan sums into a logarithmic-derivative framework and relies on Ramanujan's identity, equivalent to the prime-number theorem, to force the logarithm to vanish. Beyond the main result, the authors derive new infinite-product identities involving cyclotomic polynomials and connect these products to the Jacobi theta function, revealing modular-form structure on $\Gamma_0(4)$. A concrete consequence is a logarithmic identity expressing $\pi/4$ as a ratio of sums linked to cyclotomic logs and Ramanujan sums, illustrating deep ties between cyclotomic products, analytic number theory, and modular forms.
Abstract
We show that the infinite product defined by \[ P(z) = -\prod_{n=1}^{\infty} (Φ_n(z))^{-1/n}, \] where \( Φ_n(z) \) 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.