Note on the Odd Perfect Numbers

EasyChair Preprint 8121, version 3

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 $\frac{1}{2}$. We state the conjecture that $\frac{\pi^2}{6.4} \times e^{0.0712132519795} \times \log x \geq e^{\gamma} \times \log(x - K \times \sqrt{x})$ is satisfied for infinitely many natural numbers $x > 10^{8}$ where $K > 0$ is a constant. Under the assumption of this conjecture and the Riemann Hypothesis, we prove that there is not any odd perfect number at all.

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

@booklet{EasyChair:8121,
  author    = {Frank Vega},
  title     = {Note on the Odd Perfect Numbers},
  howpublished = {EasyChair Preprint 8121},
  year      = {EasyChair, 2022}}