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.

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

${\displaystyle 0<a_{1}<a_{2}\cdots <a_{\phi (n)}<n,}$

the mean square gap satisfies

${\displaystyle \sum _{i=1}^{\phi (n)-1}(a_{i+1}-a_{i})^{2}<Cn^{2}/\phi (n)}$

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