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 $n$. 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.