Download PDFOpen PDF in browserCurrent versionRiemann Hypothesis on Grönwall's FunctionEasyChair Preprint 9117, version 146 pages•Date: July 3, 2023AbstractGrönwall's function $G$ is defined for all natural numbers $n>1$ by $G(n)=\frac{\sigma(n)}{n \cdot \log \log n}$ where $\sigma(n)$ is the sum of the divisors of $n$ and $\log$ is the natural logarithm. We require the properties of colossally abundant numbers in relation to the Grönwall's function $G$. There are several statements equivalent to the famous Riemann hypothesis. We state that the Riemann hypothesis is true if and only if there exist infinitely many pairs $(N,N')$ of consecutive colossally abundant numbers $N< N'$ such that $G(N)< G(N')$. Using this new criterion, we prove that the Riemann hypothesis is true. Keyphrases: Arithmetic Functions, Colossally abundant numbers, Extremely abundant numbers, Hyper abundant numbers, Riemann hypothesis
