Question 71
Let $T$ be a given positive integer. Find all functions $f : \mathbb{Z} \to \mathbb{C}$ such that for every integer $m$, $f(m+T) = f(m)$, and for every three integers $a, b$, and $c$, the following equality holds:
$$
f(a)\overline{f(a+b)}f(a+c)\overline{f(a+b+c)} = 1.
$$
Note: $\bar{z}$ denotes the complex conjugate of the complex number $z$.
$$
f(a)\overline{f(a+b)}f(a+c)\overline{f(a+b+c)} = 1.
$$
Note: $\bar{z}$ denotes the complex conjugate of the complex number $z$.