Riemann Hypothesis on Grönwall's Function

EasyChair Preprint no. 9117, version 14

Abstract

Grönwall'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

@Booklet{EasyChair:9117,
  author = {Frank Vega},
  title = {Riemann Hypothesis on Grönwall's Function},
  howpublished = {EasyChair Preprint no. 9117},

  year = {EasyChair, 2023}}