Totative

In number theory, a totative of a given positive integer $n$ is an integer $k$ such that $0 < k ≤ n$ and $k$ is coprime to $n$. Euler's totient function φ(n) counts the number of totatives of n. The totatives under multiplication modulo n form the multiplicative group of integers modulo n.

Distribution
The distribution of totatives has been a subject of study. Paul Erdős conjectured that, writing the totatives of n as


 * $$ 0 < a_1 < a_2 \cdots < a_{\phi(n)} < n ,$$

the mean square gap satisfies


 * $$ \sum_{i=1}^{\phi(n)-1} (a_{i+1}-a_i)^2 < C n^2 / \phi(n) $$

for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.