The Magic of Prime Numbers

EasyChair Preprint 11786, version 3

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 $n$. 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 and the Cramér's conjecture is false. In this note, we show that the previous inequality always holds for all large enough prime numbers.

Keyphrases: Chebyshev function, Cramér's conjecture, Riemann hypothesis, Riemann zeta function, prime numbers

@booklet{EasyChair:11786,
  author    = {Frank Vega},
  title     = {The Magic of Prime Numbers},
  howpublished = {EasyChair Preprint 11786},
  year      = {EasyChair, 2024}}