Note for the Riemann Hypothesis

EasyChair Preprint 13682, version 2

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 show if the inequality $R(N_{n+1}) < R(N_{n})$ holds for $n$ big enough, then the Riemann hypothesis is true. In this note, we prove that $R(N_{n+1}) < R(N_{n})$ always holds for $n$ big enough.

Keyphrases: Chebyshev function, Riemann hypothesis, Riemann zeta function, prime numbers

@booklet{EasyChair:13682,
  author    = {Frank Vega},
  title     = {Note for the Riemann Hypothesis},
  howpublished = {EasyChair Preprint 13682},
  year      = {EasyChair, 2024}}