Simple Proof for the Riemann Hypothesis

EasyChair Preprint no. 11372

7 pagesDate: November 22, 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 a natural number $x > 1$. The Riemann hypothesis is the assertion that all non-trivial zeros are 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. In 2011, Sol{\'e} and Planat stated that the Riemann hypothesis is true if and only if the inequality $R(M_{x}) > \frac{e^{\gamma}}{\zeta(2)}$ holds for all $x \geq 5$, where $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\zeta(x)$ is the Riemann zeta function. In this note, using Sol{\'e} and Planat criterion, we prove that the Riemann hypothesis is true.

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