In this post I show: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Wolfram Alpha shows an alternate form for the primes: $$\frac{p_n{}^2}{p_n{}^2-1}=\frac{\left(\sum _{k=1}^{2^n} \left\lfloor \left(\frac{n}{\pi (k)+1}\right)^{1/n}\right\rfloor +1\right){}^2}{\left(\sum _{k=1}^{2^n} \left\lfloor \left(\frac{n}{\pi (k)+1}\right)^{1/n}\right\rfloor +1\right){}^2-1}, $$ and I note the prime counting function. I read somewhere that $\zeta(1)$ is equivalent to the Prime Number Theorem, so I have this question.
Would using primes to prove RH be circular? Or, is the fact that $s=1$ is a pole in the complex plane, mean there is no circularity?
Both identities can be proved directly, without reference to the prime number theorem or the Riemann hypothesis or each other. So there's no circularity.