Elementary totient-ratio lower bound ($\phi (r)/r\ge 1/(1+{\log }_{2}r)$)

List the $k$ distinct prime divisors of $r$ in increasing order. The $j$th is at least $j+1$, and $k\le {\log }_{2}r$. The Euler totient function product formula gives $\phi (r)/r\ge {\prod }_{j=1}^{k}j/(j+1)=1/(k+1)$. This elementary bound is sufficient to show inverse-polynomial success probability for coprime phase labels in quantum order finding.