Download PDFOpen PDF in browserCurrent versionThe Riemann HypothesisEasyChair Preprint 3708, version 49Versions: 12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152→history 5 pages•Date: May 11, 2021AbstractLet's define $\delta(x) = (\sum_{{q\leq x}}{\frac{1}{q}}\log \log xB)$, where $B \approx 0.2614972128$ is the MeisselMertens 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. A theorem due to Erhard Schmidt implies that $S(x)$ changes sign infinitely often. Using the Nicolas theorem, we prove that when the inequalities $\delta(x) \leq 0$ and $S(x) \geq 0$ are satisfied for some $x \geq 127$, then the Riemann Hypothesis should be false. However, the Mertens second theorem states that $\lim_{{x\to \infty }} \delta(x) = 0$. Moreover, a result from the Gr\"{o}nwall paper could be restated as $\lim_{{x\to \infty }} S(x) = 0$. In this way, this work could mean a new step forward in the direction for finally solving the Riemann Hypothesis. Keyphrases: Divisor, inequality, number theory
