I came across this paper from 2016 and it struck me because I tried this exact same thing in 2015 explained in this question. I know now that I was wrong because of a flaw in my thinking (I don't know where the error in the paper is in Mladen's work, but I know there has to be a mistake so if you can spot it that would be terrific). However, I still find the result, $$P(s)\approx1-\sqrt{\frac 2{\zeta(2^0s)}-\sqrt{\frac 2{\zeta(2^1s)}-\sqrt{\frac 2{\zeta(2^2s)}-\cdots-\sqrt{\lim\limits_{n \to \infty}\frac 2{\zeta(2^ns)}-1}}}}$$ approximating the prime zeta function, to be quite interesting (Mladen's equation excludes the minus one at the right end of the nested radical).
How good exactly is this approximation?
You're right, the result must be wrong. If the proof of the main result, $(1 − P(s))^2 =2/\zeta(s)− 1 + P(2s),$ were correct, it would be valid for all $s>1,$ not just integers. That's impossible, since the LHS grows to infinity as $s\rightarrow1+,$ while the RHS $\rightarrow-1+P(2).$ That's because $\zeta(s)$ has a pole at $s=1,$ and $P(s)$ a logarithmic singularity.
Numerically, the difference between both sides is small for large $s,$ predictably, as both approach $1$. For $s=2,$ it's $\approx0.007185545379369163,$ not even a satisfactory approximation. For $s=1.5,$ it's $\approx0.08228197886084833$, for $s=1.25,$ it's $\approx0.4091107555146472$.