The Holy Grail of Mathematics

EasyChair Preprint no. 8888, version 1

Versions: 12history
6 pagesDate: September 28, 2022

Abstract

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 $\frac{1}{2}$. Let $Q$ be the set of prime numbers $q_{n}$ satisfying the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \cdot \log\theta(q_{n})$ with the product extending over all prime numbers $q$ that are less than or equal to $q_{n}$, where $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\theta(x)$ is the Chebyshev function and $\log$ is the natural logarithm. If the Riemann hypothesis is false, then there are infinitely many prime numbers $q_{n}$ outside and inside of $Q$. In this note, we obtain a contradiction when we assume that there are infinitely many prime numbers $q_{n}$ outside of $Q$. By reductio ad absurdum, we prove that the Riemann hypothesis is true.

Keyphrases: Chebyshev function, Nicolas inequality, prime numbers, Riemann hypothesis