title: Totative
text: # 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
  
      $\sum _{i=1}^{\phi (n)-1}(a_{i+1}-a_{i})^{2}<Cn^{2}/\phi (n)$
  
  for some constant C, and this was proven by Bob Vaughan and Hugh Montgomery.
  
  