Classes of exponent

Question

I found this terminology in a paper (link) and did not understand it's meaning. Here is the set of lines that I am talking about: For each prime $p \le g$, we remove all residue classes $\mod p$ except the zero class and the classes of exponent $\le x$. Since there are $\phi(f)$ classes of exponent $f$ for $f | p-1$ we have $$g(p) = 1 + \sum \limits_{\substack{f \le x \\ f | p-1}} \phi(f).$$ I am assuming that $\phi(f)$ denotes the totient function but beyond that, I fail to understand the meaning of the equation and why it should be true. $g(p)$ in the context of the paper should be the number of residue classes $\mod p$.

Answer 1

The "class of exponent $k$ modulo $p$" is the collection of elements of multiplicative order $k$, except for the zero class, which is the collection of zero divisors in $\Bbb{Z}/p\Bbb{Z}$. Note that the multiplicative order of an element divides $p-1$ (since it is assumed in the statement of theorem 2 that $p$ is prime), so the set of classes have exponents dividing $p-1$.