An Equivalent to the Riemann Hypothesis

EasyChair Preprint no. 11528, version 5

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7 pagesDate: December 19, 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. There are several statements equivalent to the famous Riemann hypothesis. Robin's criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma} \cdot n \cdot \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant and $\log$ is the natural logarithm. We prove that the Riemann hypothesis is true whenever there exists a large enough positive number $x_{0}$ such that for all $x > x_{0}$ we obtain that the value of
$\sum_{n \leq \alpha_{x}} \frac{1}{n} - \sum_{6 \leq n \leq \frac{x}{\log x}} \frac{e^{-\gamma}}{n \cdot \left(\log(n \cdot \log n)\right)} - \sum_{n < 6} \frac{e^{-\gamma}}{q_{n}}$
is lesser than or equal to $e^{-\gamma} \cdot \left(\gamma - B - \frac{1}{2 \cdot (x - 1)}\right)$ where $B \approx 0.26149$ is the Meissel-Mertens constant and $\alpha_{x} = \left(\log x + \frac{0.0222 \cdot \log x}{\log \log x}\right)$. Since the previous expression goes to $0$ as $x$ tends to infinity, then we deduce that the Riemann hypothesis must be true.

Keyphrases: prime numbers, Riemann hypothesis, Robin's criterion, Superabundant numbers