Download PDFOpen PDF in browserAn Equivalent to the Riemann HypothesisEasyChair Preprint no. 11528, version 57 pages•Date: December 19, 2023AbstractThe 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 sumofdivisors function of $n$, $\gamma \approx 0.57721$ is the EulerMascheroni 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 Keyphrases: prime numbers, Riemann hypothesis, Robin's criterion, Superabundant numbers
