We have that $\phi(n)$ is the number of positive integers less than $n$ and relatively prime to $n$.
I tried to extend this definition by working with $\phi_x(n)$, which is the number of positive integer less than $x$ and relatively prime to $n$. I wonder if there was a research about this extension before, because I found a very interesting inequality. (haven't proved yet, but I checked for some value of $m,n$)
Let $m,n$ be positive integers. Then we have:
$\phi(1)+\phi(2)+\cdots+\phi(m)+\phi_n(1)+\phi_n(2)+\cdots+\phi_n(m) \geq \phi(n+1)+\phi(n+2)+\cdots+\phi(n+m)$
Does anyone have any hint for this or any resources related to $\phi_x(n)$? Thank you in advance.