Note for the Prime Numbers

Note for the Prime Numbers

Postby Guest » Tue Mar 26, 2024 11:13 am

I would be very grateful if you as an expert in the field would consider reviewing this publication:

https://www.scienceopen.com/document/re ... c65021cedb

To review, go to that page and click on the “Review” button or consult the details about the reviewing process at ScienceOpen

Thank you very much also for your cooperation. I look forward to your review!

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The following are some hints to help you with the revision:
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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 $n$. We prove that if the inequality $R(N_{n+1}) < R(N_{n})$ holds for all primes $q_{n}$ (greater than some threshold), then the Riemann hypothesis is true and the Cramér's conjecture is false. In this note, we show that the previous inequality always holds for all large enough prime numbers. Indeed, the manuscript only pretends to be a proof of the Riemann hypothesis. However, it is known that the inequality $R(N_{n+1}) < R(N_{n})$ should be false for always large enough natural numbers $n$ under the assumption that the Cramér's conjecture is true (we show the inequality holds indeed for large $n$ and thus, Cramér's conjecture must be false). This argument belongs to a paper written by well-known number theorists:

https://arxiv.org/abs/1012.3613

Thanks in advance :smile:
Guest
 

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