Riemann Hypothesis on Grönwall's Function

EasyChair Preprint no. 9117, version 7

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 extremely abundant numbers, that is to say left to right maxima of $n \mapsto G(n)$. We also use the colossally abundant and hyper abundant numbers. 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 consecutive colossally abundant numbers $N< N'$ such that $G(N)< G(N')$. In addition, we prove that the Riemann hypothesis is true when there exist infinitely many hyper abundant numbers $n$ with any parameter $u > 1$. We claim that there could be infinitely many hyper abundant numbers with any parameter $u > 1$ and thus, the Riemann hypothesis would be 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}}