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.

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 proved by Bob Vaughan and Hugh Montgomery.^[1]

References

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