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Note for the Prime Numbers
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 $n$. On the one hand, the Cramér's conjecture is an estimate for the size of gaps between consecutive prime numbers. This was formulated more than eighty years ago. On the other hand, a correct proof for the Riemann hypothesis has been considered as the Holy Grail of Mathematics. The Riemann hypothesis is concerned with the locations of the nontrivial zeros of the Riemann zeta function. This remains open since more than one hundred and sixty years ago. There are several statements equivalent to the famous Riemann hypothesis. 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.
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References
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