Tight bounds for $\Phi_n(b)$?

Question

What are tight bounds for $$\Phi_n(b)$$ where $\ \Phi_n()\ $ denotes the $\ n $-th cyclotomic polynomial , $\ n\ge 3\ $ and $\ b\ $ is a "large" positive integer ?

We can assume that the prime factorization of $\ n\ $ is known and hence $\ \varphi(n)\ $. $$b^{\varphi(n)}$$ is the obvious approximation and seems to be already quite good , in particular if $\ n\ $ is large as well. But this approximation can be too small or too big, I guess depending on $\ n\ $, but I could not find out yet how. Any reference or ideas?

Answer 1

From this document titled The cyclotomic polynomials - Notes by G.J.O. Jameson, we have from page $2$, section 1.2 (v), the following proposition:

For all $n \geq 3$, and for all $b > 0$, we have $$(b - 1)^{\varphi(n)} < \Phi_n(b) < (b + 1)^{\varphi(n)}.$$

I am not too sure, though, if these bounds are tight for large $b$, as I have just commenced my self-study of cyclotomic polynomials.