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Note for the Prime Numbers
Revision notes
We removed the result that the Cramér's conjecture is false. We considered this result only as a possible extension of this work since the used reference has never been published by a journal under peer-review (it is only available in arXiv despite of it was written by well-known number theorists). Indeed, we added a Discussion section to include it. For that reason, we changed the Abstract and Keywords.
Abstract
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 prove 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. Moreover, we discuss some known implications about this inequality and the prime numbers. In this note, we show that the previous inequality always holds for all large enough prime numbers.
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References
Connes Alain. An Essay on the Riemann HypothesisOpen Problems in Mathematics. p. 225–257. 2016. Springer International Publishing. [Cross Ref]
Euler's Totient FunctionEquivalents of the Riemann Hypothesis. p. 94–143. Cambridge University Press. [Cross Ref]
Ayoub Raymond. Euler and the Zeta Function. The American Mathematical Monthly. Vol. 81(10)1974. JSTOR. [Cross Ref]
Nicolas Jean-Louis. Petites valeurs de la fonction d'Euler. Journal of Number Theory. Vol. 17(3):375–388. 1983. Elsevier BV. [Cross Ref]