Totative

In number theory, a totative of a given positive integer $n$ is an integer $k$ such that $0 < k \leq n$ 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} \dots < a_{ϕ (n)} < n,

the mean square gap satisfies

\sum_{i = 1}^{ϕ (n) - 1} (a_{i + 1} - a_{i})^{2} < C n^{2} / ϕ (n)

for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.^[1]

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totative at PlanetMath.

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