Is this a route to the prime number theorem?

[arivero]

I notice that the trick to define Dirichlet Eta Function can be repeated for each prime number, let p a prime, and then

$$\eta_p= (1 - p^{-s}) \zeta(s) - p^{-s} \zeta(s) = (1 - 2 p^s) \zeta(s)$$

So each prime p defines a function $$\eta_p$$ that adds a family of zeros at $s= (\log 2 + 2 n i \pi) / \log p$. Particularly, for p=2 it kills the only pole of zeta in s = 1.

Repeating the trick for each prime, we seem to obtain a function $$\eta_\infty(s)$$ having the same zeros that the Riemann zeta plus families of zeros of density 1/log p placed at the lines r=log(2)/log(p)

Or, if we don't like to crowd the critical strip, we can use only the left part of the eta, the $$\eta^F_p(s)= (1 - p^{-s}) \zeta(s)$$, and then for $$\eta^F_\infty(s)$$ we get all the families cummulated in the r=0 line.

But on other hand it can be argued that $$\eta^F_\infty(s)=1$$, as we have removed all the factors in Euler product. So there should be some relation between the n/log(p) zeros in the imaginary line and the other zeros in the Riemann function

 

[fresh_42]

It is hard to tell anything as long as $\eta_\infty$ is basically undefined.