Inspired by this mysterious function :
$f(x) + f(x/2) + f(x/3) + f(x/4) + ... = x$ and $\lim_{n \to \infty} \frac{f(n)}{\pi(n)} = 1$?
I started to wonder since the alternating sum equals zero :
$$f(x) - f(x/2) + f(x/3) - f(x/4) + ... = 0$$
Do we have for every solution
$$t(x) - t(x/2) + t(x/3) - t(x/4) + ... = 0$$
the implication
$$t(x) + t(x/2)+t(x/3)+t(x/4) = C x$$
For some constant $C$ ??