Counterexample of the Riemann Hypothesis

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 $\prod_{q \leq x} \frac{q}{q-1} < \left(e^{\gamma} \times \log x\right) \times \left(1 - \frac{\log x}{8 \times \pi \times \sqrt{x}}\right)$ for every $x \geq 599$. Hence, we obtain that if the Riemann hypothesis is true, then $x^{\left(1 - \frac{\log x}{8 \times \pi \times \sqrt{x}}\right)} > \theta(x)$ for every $x \geq 599$. However, this is false since $(\theta(x) - x)$ changes sign infinitely often. By contraposition, the Riemann hypothesis is false.