In Table 5.1 on page 103 of the book The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators by Machiel van Frankenhuijsen the author states:
$$\textit{Riemann hypothesis} \iff \psi(x)=x+O(x^{1/2+\epsilon}) \iff \psi(x) \le x+O(x^{1/2+\epsilon})$$
where $\psi(x) = \sum_{k \in \mathbb{N}} \sum_{p^k \le x} \log(p)$ is the second Chebyshev function.
He also writes (a page earlier):
As a consequence of the density in the real line of the values $\log \left\lvert x \right\rvert$ $(x \in \mathbb{Q})$, it becomes unnecessary to establish the lower bound ...
He leaves it at that, however, and neither have I found this worked out somewhere nor have I been able to fill in the details. So
Question: How do you proof that an upper bound is enough? That is, how do you use the density of $\log \left\lvert x \right\rvert$ $(x \in \mathbb{Q})$ in $\mathbb{R}$ to turn the upper bound into a lower bound?