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. Let's also define $S(x) = \theta(x) - x$, where $\theta(x)$ is the Chebyshev function. It is known that $S(x)$ changes sign infinitely often. We define the another function $\varpi(x) = \left(\sum_{{q\leq x}}{\frac{1}{q}}-\log \log \theta(x)-B \right)$. We prove that when the inequality $\varpi(x) \leq 0$ is satisfied for some number $x \geq 3$, then the Riemann Hypothesis should be false. The Riemann Hypothesis is also false when the inequalities $\delta(x) \leq 0$ and $S(x)\geq 0$ are satisfied for some number $x \geq 3$ or when $\int_{x}^{\infty} \frac{S(y) \times (1 + \log y)}{y^{2} \times \log^{2} y} dy \geq \frac{S(x)^{2}}{x^{2} \times \log x}$ is satisfied for some number $x \geq 121$.