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This question is motivated by this answer.

The $n$-th cyclotomic polynomial $\Phi_n(x)$ has degree $\phi(n)$ and so behaves like $x^{\phi(n)}$ for $x$ large.

I'm interested in a more precise growth.

When $\Phi_n(x)$ has positive coefficients, it is obvious that $\Phi_n(x) > x^{\phi(n)}$ when $x > 0$. Unfortunately, $\Phi_n(x)$ can have negative coefficients. As a consequence, for several $n$, we have $\Phi_n(x) > x^{\phi(n)}$ only for $x<1$. This happens for $n=6$ for instance.

So, I'm left with this:

When does $\Phi_n(x) > x^{\phi(n)-1}$ for $x>1$ ?

Because of the motivation mentioned above, I'm interested in $x=5$, but it'd be interesting to know finite lower bounds for which this inequality holds.

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  • Have you looked at this result? https://math.stackexchange.com/questions/880395/lower-bound-for-cyclotomic-polynomials-evaluated-at-an-integer – j4l3kl24jkl2 Apr 06 '17 at 16:06
  • @Tbraz, thanks! $\Phi_n(a)\gt a^{\frac{\phi(n)}{2}}$ is nice but unfortunately is not enough to solve the motivating question. – lhf Apr 06 '17 at 16:11
  • http://mathoverflow.net/questions/221357/cyclotomic-polynomials-phi-np-is-like-p-phin-for-big-enough-p-ri seems relavant – lhf Apr 07 '17 at 13:36

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