New Criterion for the Riemann Hypothesis

EasyChair Preprint no. 11415, version 1

8 pagesDate: November 29, 2023

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. 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 $n$. There are several statements equivalent to the Riemann hypothesis. We state that if for each large enough prime number $q_{n}$, there exists another prime $q_{n'} > q_{n}$ such that $R(N_{n'}) \leq R(N_{n})$, then the Riemann hypothesis is true. In this note, using our criterion, we prove that the Riemann hypothesis is true.

Keyphrases: Chebyshev function, elementary number theory, prime numbers, Riemann hypothesis, Riemann zeta function