preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202002.0379.v11

Preprint Article Version 11 Preserved in Portico This version is not peer-reviewed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)
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Version 8 : Received: 2 July 2021 / Approved: 6 July 2021 / Online: 6 July 2021 (12:38:05 CEST)
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Version 10 : Received: 20 June 2024 / Approved: 21 June 2024 / Online: 21 June 2024 (10:43:53 CEST)
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Version 12 : Received: 25 June 2024 / Approved: 25 June 2024 / Online: 25 June 2024 (10:26:09 CEST)

Let $\Psi(n) = n \cdot \prod_{q \mid n} \left(1 + \frac{1}{q} \right)$ denote the Dedekind $\Psi$ function where $q \mid n$ means the prime $q$ divides $n$. Define, for $n \geq 3$; the ratio $R(n) = \frac{\Psi(n)}{n \cdot \log \log n}$ where $\log$ is the natural logarithm. Let $N_{n} = 2 \cdot \ldots \cdot q_{n}$ be the primorial of order $n$. A trustworthy proof for the Riemann hypothesis has been considered as the Holy Grail of Mathematics by several authors. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. There are several statements equivalent to the famous Riemann hypothesis. We show if the inequality $R(N_{n+1}) < R(N_{n})$ holds for $n$ big enough, then the Riemann hypothesis is true. In this note, we prove that $R(N_{n+1}) < R(N_{n})$ always holds for $n$ big enough.

Riemann hypothesis; Prime numbers; Riemann zeta function; Chebyshev function

Computer Science and Mathematics, Algebra and Number Theory

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