Let $\pi(x)$ the prime-counting function and $\varphi(n)$ denotes the Euler's totient function.
I would like to know if next arithmetic function, that I've defined searching a comparison of such arithmetic function, tends to zero as $N\to\infty$
When $N=10^D$, with $D=0,1,2,3$ one gets that $f(N)$ is calculated as, respectively, $1,\approx0.917,\approx0.511$ and $\approx0.370$. Thus our positive function is decreasing in this segment.
Question. Prove or refute $$f(N)\to 0$$ as $N\to\infty$. Many thanks.
I presume that the solution needs the prime number theorem and inequalities for the Euler's totient function.