On inequalities involving counts of the prime factors of an odd perfect number

Abstract

Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number (counting multiplicity) of prime factors of $N$. We prove that $\frac{99}{37}\omega(N) - \frac{187}{37} \leq \Omega(N)$ and that if $3\nmid N$, then $\frac{51}{19}\omega(N)-\frac{46}{19} \leq \Omega(N)$.