Rosser–Schoenfeld totient bound ($\phi (n)/n>1/[e^{{\gamma }_E}\log \log n+2.50637/\log \log n]$)

For $n\ge 3$, this explicit lower bound on the Euler totient function implies $\phi (n)/n=\Omega (1/\log \log n)$ as $n\to \infty$. Here ${\gamma }_E$ is the Euler--Mascheroni constant. The prime-product formula also gives $\phi (r)/r\ge \phi (N)/N$ whenever $r\mid N$, making the bound useful for certified quantum period finding. The relevant statement is a lower bound: the claimed universal upper estimate $\phi (n)=O(n/\log \log n)$ would fail at primes. Equations (3.41)–(3.42) of Theorem 15 provide the primary bound: denisevellachemla.eu/Rosser-Schoenfeld-1962.pdf .