New techniques for bounds on the total number of prime factors of an odd perfect number

Abstract:

Let $\sigma (n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\sigma (n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form $N = p^\alpha \prod _{j=1}^k q_j^{2 \beta _j}$, where $p, q_1, \cdots , q_k$ are distinct primes and $p \equiv \alpha \equiv 1 \pmod {4}$. Define the total number of prime factors of $N$ as $\Omega (N) := \alpha + 2 \sum _{j=1}^k \beta _j$. Sayers showed that $\Omega (N) \geq 29$. This was later extended by Iannucci and Sorli to show that $\Omega (N) \geq 37$. This was extended by the author to show that $\Omega (N) \geq 47$. Using an idea of Carl Pomerance this paper extends these results. The current new bound is $\Omega (N) \geq 75$.