= Rosser–Schoenfeld totient bound
{c}
{title2=$\varphi(n)/n>1/[e^{\gamma_E}\log\log n+2.50637/\log\log n]$}
For $n\ge3$, this explicit lower bound on the <Euler totient function> implies $\varphi(n)/n=\Omega(1/\log\log n)$ as $n\to\infty$. Here $\gamma_E$ is the <Euler--Mascheroni constant>. The prime-product formula also gives $\varphi(r)/r\ge\varphi(N)/N$ 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 $\varphi(n)=O(n/\log\log n)$ would fail at primes. Equations (3.41)–(3.42) of Theorem 15 provide the primary bound: https://denisevellachemla.eu/Rosser-Schoenfeld-1962.pdf .
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