preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202002.0379.v8

Preprint Short Note Version 8 Preserved in Portico This version is not peer-reviewed

Version 1 : Received: 24 February 2020 / Approved: 25 February 2020 / Online: 25 February 2020 (12:21:49 CET)
Version 2 : Received: 27 February 2020 / Approved: 27 February 2020 / Online: 27 February 2020 (10:49:49 CET)
Version 3 : Received: 10 March 2020 / Approved: 11 March 2020 / Online: 11 March 2020 (16:04:28 CET)
Version 4 : Received: 31 March 2020 / Approved: 2 April 2020 / Online: 2 April 2020 (18:25:32 CEST)
Version 5 : Received: 20 April 2020 / Approved: 22 April 2020 / Online: 22 April 2020 (09:48:30 CEST)
Version 6 : Received: 3 June 2020 / Approved: 4 June 2020 / Online: 4 June 2020 (13:22:40 CEST)
Version 7 : Received: 6 June 2020 / Approved: 8 June 2020 / Online: 8 June 2020 (10:31:19 CEST)
Version 8 : Received: 2 July 2021 / Approved: 6 July 2021 / Online: 6 July 2021 (12:38:05 CEST)
Version 9 : Received: 14 October 2021 / Approved: 14 October 2021 / Online: 14 October 2021 (14:15:38 CEST)

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 $\lim_{{x \to \infty }} \frac{\theta(x)}{x} = 1$.

Riemann hypothesis; Nicolas theorem; prime numbers; Chebyshev function

Computer Science and Mathematics, Algebra and Number Theory

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