Constancy of an Infinite Cyclotomic Product via Ramanujan Sums

TL;DR

This paper proves that a specific infinite cyclotomic product remains constant within the unit disk, connecting Ramanujan's classical results on Ramanujan sums to properties of cyclotomic polynomials.

Contribution

It establishes the constancy of an infinite cyclotomic product inside the unit disk and links Ramanujan's identities to new results on such products.

Findings

The infinite product P(z) is constant inside the unit disk.

Ramanujan sums relate to the behavior of cyclotomic products.

Additional identities lead to further properties of cyclotomic products.

Abstract

We show that the infinite product defined by \[ P(z) = -\prod_{n=1}^{\infty} (\Phi_n(z))^{-1/n}, \] where \( \Phi_n(z) \) 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.