Euler's Totient function bound

Question

Do there exist any integers $n$ such that Euler's totient function $\phi(n) < n/5$? How should I approach solving this problem?

Answer 1

We know that $$\varphi(n) = n\prod_{p \mid n} \left( 1 - \frac{1}{p} \right),$$ so your question asks whether or not $$ \prod_{p \mid n} \left( 1 - \frac{1}{p} \right) < \frac{1}{5}$$ can ever hold. In fact, the limit of the product of terms of the form $\left( 1 - \frac{1}{p} \right)$ is $0$ (which is equivalent to the fact that the sum of the reciprocals of the primes diverges).

So there are infinitely many numbers $n$ such that $\varphi(n) < n/5$, and in fact infinitely many numbers $n$ such that $\varphi(n) < n/k$ for any $k$. This also makes it easy to see that the smallest number $n$ such that $\varphi(n) < n/k$ will be the product of the first $\ell$ primes for some $\ell$.

Now that we know this, it's easy to check that $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 = 30030$, and $\varphi(30030) < 30030/5$. $\diamondsuit$