On the Existence of Odd Perfect Numbers

Abstract

In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.

As pointed out by Jan-Christoph Schlage-Puchta via MO: "This only shows that there is no trivial, i.e. purely analytic, argument proving that this expression is unbounded. But there might be only finitely many odd perfect numbers, or there might be some relation between n and q, which implies that the ratio is bounded."