I was trying to create a formula for the Prime Zeta function and I partially succeeded except for one frustrating error. I was only able to formulate an approximation.
Consider the following sum: $$f_k(s)=\sum_{q\space\nmid\space{p_{n\le k}}}^\infty \frac 1{q^s}$$
This is the infinite sum of the reciprocals of all the numbers, raised to the power s, that are not divisible by any primes less than or equal to the kth prime.
For example, $f_1(s)=1+\frac 1{3^s}+\frac 1{5^s}+\frac 1{7^s}+\frac 1{9^s}+\frac 1{11^s}+\frac 1{13^s}+\frac 1{15^s}+\cdots$
$\qquad\qquad\quad$ $f_2(s)=1+\frac 1{5^s}+\frac 1{7^s}+\frac 1{11^s}+\frac 1{13^s}+\frac 1{17^s}+\frac 1{19^s}+\frac 1{23^s}+\frac 1{25^s}+\cdots$
By taking $f_k(s)-({p_{k+1}}^{-s})f_k(s)$ we end up with $f_{k+1}(s)$. In other words $$f_k(s)\left(\frac {{p_{k+1}}^s-1}{{p_{k+1}}^s}\right)=f_{k+1}(s)$$
Therefore, since we define $f_0(s)=\zeta(s)$, it's clear that $$f_k(s)=\zeta(s)\left(\prod_{n=1}^k\frac {{p_n}^s-1}{{p_n}^s}\right)$$
It can also be shown that $$f_k(s)=\zeta(s)\left(\frac {P(k,s)^2-P(k,2s)}2-P(k,s)+1\right)$$ where $P(k,s)$ is the partial Prime Zeta Function: $$P(k,s)=\sum_{p\in primes}^k \frac 1{p^s}$$
This is where I hit a snag. I want to equate the above two equations for $f_k(s)$ like this: $$f_k(s)=\zeta(s)\left(\prod_{n=1}^k\frac {{p_n}^s-1}{{p_n}^s}\right)=\zeta(s)\left(\frac {P(k,s)^2-P(k,2s)}2-P(k,s)+1\right)$$ In theory they should be equal, but when I test them, they show to be slightly different, especially for larger k. Why is this? I can't figure out a single reason why they should be unequal.
If they were perfectly equal, then $$P(s)=\lim\limits_{x \to \infty} \left(1-\sqrt{\frac 2{\zeta(2^0s)}-\sqrt{\frac 2{\zeta(2^1s)}-\sqrt{\frac 2{\zeta(2^2s)}-\cdots-\sqrt{\frac 2{\zeta(2^xs)}-1}}}}\right)$$ However, they are only almost equal and thus this is simply an approximation.
The "it can also be shown" expression is wrong. It only contains terms with up to two different prime factors. Write it out for $k=3$ to see that it differs from the correct expression by a term $\zeta(s)(2\cdot3\cdot5)^{-s}$.