# Note for the Prime Numbers

### Note for the Prime Numbers

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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