Riemann Hypothesis on Odd Perfect Numbers

EasyChair Preprint no. 9029

6 pagesDate: October 8, 2022

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 $\frac{1}{2}$. The Riemann hypothesis is considered by many to be the most important unsolved problem in pure mathematics. Let $\sigma(n)$ denote the sum-of-divisors function $\sigma(n)=\sum_{d \mid n} d$. An integer $n$ is perfect if $\sigma(n)=2 \cdot n$. It is unknown whether any odd perfect numbers exist. Leonhard Euler stated: Whether $\ldots$ there are any odd perfect numbers is a most difficult question''. We require the properties of superabundant numbers, that is to say left to right maxima of $n \mapsto \frac{\sigma(n)}{n}$. We also use Robin's criterion and Ramanujan's old notes which were published in 1997 annotated by Jean-Louis Nicolas and Guy Robin. There are several statements equivalent to the famous Riemann hypothesis. In this note, conditional on Riemann hypothesis, we prove that there is no odd perfect number.

Keyphrases: Odd perfect numbers, prime numbers, Riemann hypothesis, sum-of-divisors function, Superabundant numbers