Riemann Hypothesis on Superabundant Numbers

EasyChair Preprint no. 11515, version 2

Versions: 12history
10 pagesDate: December 13, 2023

Abstract

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. Let $\Psi(n) = n \cdot \prod_{q \mid n} \left(1 + \frac{1}{q} \right)$ denote the Dedekind $\Psi$ function where $q \mid n$ means the prime $q$ divides $n$. Define, for $n \geq 3$; the ratio $R(n) = \frac{\Psi(n)}{n \cdot \log \log n}$ where $\log$ is the natural logarithm. Let $\sigma(n)$ denote the sum-of-divisors function $\sigma(n)=\sum_{d \mid n} d$. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. There are several statements equivalent to the Riemann hypothesis. If for each large enough superabundant number $n$, there exists another superabundant $n' > n$ such that $R(n') \leq R(n)$, then the Riemann hypothesis is true. In this note, using this criterion on superabundant numbers, we prove that the Riemann hypothesis is true.

Keyphrases: Chebyshev function, prime numbers, Riemann hypothesis, Riemann zeta function, Superabundant numbers