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 $\frac{3\times\log x+5}{8\times\pi\times\sqrt{x}+1.2\times\log x+2}+\frac{\log x}{\log\theta(x)}\leq 1$ is satisfied for some number $x\geq 13.1$ or when there exists some number $y\geq 13.1$ such that for all $x\geq y$ the inequality $\frac{3\times\log x+5}{8\times \pi \times\sqrt{x}+1.2 \times\log x+2}+\frac{\log x}{\log(x+C \times\sqrt{x} \times \log\log\log x)}\leq 1$ is always satisfied for some positive constant $C$ independent of $x$.