Let $a$ be an integer, a>1 such that is not the $d-th$ power of any integer for $d$ dividin $n$ and $d>1$. Let $f:\mathbb{T}^n\rightarrow\mathbb{T}^n$ a continius function such that the Lefschetz´s numbers of $f$ are given by
$$L(f^m)=(a^{\frac{m}{gcd(m,n)}}-1)^{gcd(m,n)}$$
The Lefschetz´s numbers of period m is defined by:
$$l(f^m)=\sum_{r|m}\mu(r)L(f^{m/r})$$
I must to proof that $l(f^m)\neq 0$
My Aproach With the help of Mathematica I have seen that for $a=2$ , $n=24$ then $l(f^m)=0$ for each $m$ dividing 24. Otherwise, also using Mathematica I have seen some values of $l(f^m)$ for diferent $n$ and its values are in fact different from $0$. When $m$ is prime the problem it is easy. Using Mathematica I have seen again and again that $L(f^m)$ is is so much bigger than the other $L(f^{m/r})$ and that is why $l(f^{m})\neq 0$ for $a>2$
Does anybody has a hint?