Elementary Proof of No Odd Perfect Numbers

Question

I came across this proof on the Arxiv that there are no odd perfect numbers. It is elementary and easy to follow and looks correct to me? Of course there must be a mistake there somewhere but I am not catching it - could someone else take a look?

Answer 1

I see (3.6) is wrong:

The third term, $\prod _{i=1} ^ m p_i \sigma (p_i ^ {a_i-1})$ is as one expects, the rest. It includes those divisors that have at least one $p_1$ but also include some other prime or primes.

I don't think this is right, there's no way from this we could get, say, $p_1p_2$ (with $m \ge 3$) since $p_3$ is always in the product.