Numerical Computations on Prime Numbers

EasyChair Preprint 11752, version 1

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 state 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 Cram{\'e}r's conjecture is false. In this note, we prove that the previous inequality always holds for all sufficiently large primes $q_{n}$. This manuscript was based on numerical computations which confirm and support the truthfulness of this mathematical result.

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

@booklet{EasyChair:11752,
  author    = {Frank Vega},
  title     = {Numerical Computations on Prime Numbers},
  howpublished = {EasyChair Preprint 11752},
  year      = {EasyChair, 2024}}