The cyclotomic prime-divisor sparsity conjecture

Let $\ell$ be an odd prime, let $\Phi_\ell(X)$ be the $\ell$th cyclotomic polynomial, and let $\mathbb{P}$ be the set of primes dividing some value $\Phi_\ell(d)$ with $d\in\mathbb{N}$. For a positive integer $i$ and a real parameter $T$, define

$$R_\ell(n)=\min\{d\in\mathbb{N}:n\mid\Phi_\ell(d)\},$$

when such a $d$ exists, and set $R_\ell(n)=\infty$ otherwise, and define

$$S^i(\ell,T)=\{p\in\mathbb{P}:R_\ell(p^i)\leq T\}.$$

Cyclotomic prime-divisor sparsity conjecture. One has

$$|S^2(\ell,T)|=o(T).$$

This estimate is proposed on the basis of empirical evidence and is intended to control the primes whose squares divide values of $\Phi_\ell$. Its status is open in the source.