Riemann Hypothesis on Grönwall's Function

EasyChair Preprint 9117, version 4

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. A number $n$ is said to be colossally abundant if, for some $\epsilon > 0$, $\frac{\sigma(n)}{n^{1 + \epsilon}} \geq \frac{\sigma(m)}{m^{1 + \epsilon}}$ for all $m > 1$. Let us call hyper abundant an integer $n$ for which there exists $u > 0$ such that $\frac{\sigma(n)}{n \cdot (\log n)^{u}} \geq \frac{\sigma(m)}{m \cdot (\log m)^{u}}$ for all $m > 1$. 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. We state that the Riemann hypothesis is true if and only if there exist infinitely many consecutive colossally abundant numbers $3 \leq N < N'$ such that $G(N) \leq G(N')$. In addition, we prove that the Riemann hypothesis is true when there exist infinitely many hyper abundant numbers $n$ with any parameter $u \gtrapprox 1$. We claim that there could be infinitely many hyper abundant numbers with any parameter $u \gtrapprox 1$ 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 9117},
  year      = {EasyChair, 2023}}