Note on the Odd Perfect Numbers

EasyChair Preprint no. 8121, version 2

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7 pagesDate: May 31, 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 $\frac{1}{2}$. In 2011, Sol{\'e} and and Planat stated that the Riemann Hypothesis is true if and only if $\frac{\pi^2}{6} \times \prod_{q\leq q_{n}}\left(1+\frac{1}{q} \right) >e^{\gamma}\times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 3$, where $\theta(x)$ is the Chebyshev function, $\gamma\approx 0.57721$ is the Euler-Mascheroni constant and $\log$ is the natural logarithm. We state the conjecture that $\frac{\pi^2}{6.4} \times \prod_{q \leq q_{n}} \left(1 + \frac{1}{q} \right) > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 10^{8}$. Under the assumption of this conjecture, we prove that there is not any odd perfect number at all.

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