In A conjecture concerning primes and algebra on MSE, I defined a multiplicative function $\omega:\mathbb Z_+\!\!\to\mathbb Z_+$ with $\omega(p_n)=n$, for the $n$-th prime $p_n$. It was conjectured that $\omega(N)\leq\pi(N)$ for $N>49$, where $\pi$ is the prime-counting function (which seems to be solved by a nice lemma by Greg Martin).
Now I have found another conjecture concerning $\omega$. I have tested $\varphi(N)\geq\omega(N)$ for all $N<1.000.000$, where $\varphi$ is the Euler totient function, without finding a counterexample.
The conjecture, for which I would like a proof or a counter proof, is then: $\displaystyle n\prod_{p|n}\left(1-\frac{1}{p}\right)\geq\prod_{k\geq 1}k^{n_k}$, where $\displaystyle n=\prod_{k\geq 1}p_k^{n_k}$.
Notes:
- $\varphi$ is multiplicative, $\varphi(mn)=\varphi(m)\varphi(n)$, for $(m,n)=1$
- $\omega$ is completely multiplicative, $\omega(mn)=\omega(m)\omega(n)$ for all $m,n\in\mathbb Z_+$.
- The inequality $\varphi(N)\geq\pi(N)$ should also be true, for sufficiently large $N$, but I haven't found anything about this on internet.
Jaycob Colemans comment seems to prove the inequality:
$\displaystyle \prod_{p_k,n_k\ne0}\left(1-\frac{1}{p_k}\right)\geq\prod_{p_k,n_k\ne0}\frac{k^{n_k}}{p_k^{n_k}}$, since $\displaystyle \frac{p_k-1}{p_k}\geq\left(\frac{k}{p_k}\right)^{n_k}$.