Two Major Conjectures on Prime Numbers

EasyChair Preprint 11717, version 1

Abstract

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 $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. In addition, the Cram{\'e}r's conjecture states that $q_{n+1}-q_{n}=O((\log q_{n})^{2})$, where $q_{n}$ denotes the nth prime number, $O$ is big $O$ notation, and $\log$ is the natural logarithm. 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 $R(n) = \frac{\Psi(n)}{n \cdot \log \log n}$. Let $N_{n} = 2 \cdot \ldots \cdot q_{n}$ be the primorial of order $n$. There are several statements equivalent to the Riemann hypothesis. 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 $q_{n}$.

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

@booklet{EasyChair:11717,
  author    = {Frank Vega},
  title     = {Two Major Conjectures on Prime Numbers},
  howpublished = {EasyChair Preprint 11717},
  year      = {EasyChair, 2024}}