Another Criterion For The Riemann Hypothesis

Let's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}-\log \log x-B)$, where $B \approx 0.2614972128$ is the Meissel-Mertens constant. The Robin theorem states that $\delta(x)$ changes sign infinitely often. For $x \geq 2$, Nicolas defined the function $u(x) = \sum_{q > x} \left(\log( \frac{q}{q-1}) - \frac{1}{q} \right)$ and proved that $0 < u(x) \leq \frac{1}{2 \times (x - 1)}$. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$, where $\theta(x)$ is the Chebyshev function. Using the Nicolas theorem, we demonstrate that the Riemann Hypothesis is true if and only if the inequality $\varpi(x) > u(x)$ is satisfied for all number $x \geq 3$. Consequently, we show that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann Hypothesis should be false. Moreover, if the inequalities $\delta(x) \leq 0$ and $\theta(x) \geq x$ are satisfied for some number $x \geq 3$, then the Riemann Hypothesis should be false. In addition, we know that $\lim_{{x\to \infty }} \varpi(x) = 0$ because of $\lim_{{x\to \infty }} \delta(x) = 0$ and $\lim_{{x \to \infty }} \frac{\theta(x)}{x} = 1$.