I've been working my way through the MIT OCW course on number theory, and the lecture on the PNT states:
However it was not until a century later that the prime number theorem was independently proved by Hadamard and de la Vallée Poussin in 1896, building on the work of Riemann, who in 1860 showed that there is a precise connection between the zeros of ζ(s) and the distribution of primes (we shall say more about this later), but was unable to prove the prime number theorem.
The paper cited of Riemann's is of course this one, which builds to the conclusion that the empirically known approximation $\pi(n) \sim \mathrm{Li}(n)$ has an error of $\mathrm{O}(\sqrt{n})$. Now, the presentation is a little sketchy, but I don't see any step that isn't justified, so I was wondering - why is that not considered a proof?
Anyhoo, I myself am responding to this because I realized the answer, and no one made it all that clear in the comment thread, or answered at all. Although Riemann dismissed his infamous hypothesis at the beginning, he quietly reintroduced it at the end in assuming that the $\alpha$ in $\sum^\alpha\frac{\cos(\alpha\log x)x^{-\frac{1}{2}}}{\log x}$ is real.