Define $\psi(n)=\pi(n)-\phi(n)$ where we have the prime counting function and totient function respectively.
I'm interested in where $\psi(n)=0$.
Specifically is it possible to prove that there are exactly $9$ roots, namely for $n=0,2,3,4,8,10,14,20,90$. I calculated these roots but I couldn't find any more.
I think this function has finitely many roots because the totient function grows faster and thus becomes greater than the prime counting function at a certain threshold.
Thanks for your help. I'm not sure how simple or how difficult this is to prove, and I appreciate any ideas and hints.