There's a form of complex integral what Riemann obtained to finding $\pi (x)$, $$ \pi^{*}(x)=\int_{L}\frac{\log \zeta (s)}{s}x^{s}ds, (1)$$ we already know that it lead us to the Prime Number Theorem.

But, here we can express the prime counting function with $$ \pi (x) - \pi (\sqrt x) + 1 = \int_{L} \frac{\zeta (s)\prod_{p\leq \sqrt x}\Big(1-p^{-s} \Big) }{s}x^{s}ds. (2)$$ It looks simple, but not a form to substituting it for the residues since that finite product diverges at Re$s\rightarrow -\infty$, and the behavior of the finite product is not a simple one. So, we would approximate the finite product to another function like as De Bruijn does in his paper "On the number of positive integers ≦ x and free of prime factors > y (1951.)". The paper handles inverse form of above finite product. So, what I want to ask is that, are there some papers those attempting to prove the Prime Number Theorem with (2)?