Using a Sharp bound for the Chebyshev function

Under the assumption that the Riemann hypothesis is true, von Koch deduced the asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$. A precise version of this was given by Schoenfeld. He found under the assumption that the Riemann hypothesis is true that $\left| \theta(x) - x \right| < \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. Using this result, we prove that if the Riemann hypothesis is true, then $\prod_{q \leq x} \frac{q}{q-1} > \left(e^{\gamma} \times \log x\right) \times \left(1 + \frac{\log (1 - \frac{1}{8 \times \pi \times \sqrt{x}} \times \log^{2} x)}{\log x}\right)$ for every $x \geq 599$.