Criterion for the Riemann Hypothesis

EasyChair Preprint no. 11299, version 4

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10 pagesDate: November 25, 2023

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 $M_{x} = \prod_{q \leq x} q$ be the product extending over all prime numbers $q$ that are less than or equal to $x \geq 2$. 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. There are several statements equivalent to the Riemann hypothesis. We state that if the Riemann hypothesis is false, then there exist infinitely many natural numbers $x$ such that the inequality $R(M_{x}) < \frac{e^{\gamma}}{\zeta(2)}$ holds, where $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\zeta(x)$ is the Riemann zeta function. In this note, using our criterion, we prove that the Riemann hypothesis is true.

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