Roy Lewis, Jr.

In this article, we prove the limit formula \lim_{x \to \infty} \frac{|M(x)|}{\pi(x)} = \lim_{x \to \infty} \frac{h}{log(x)} = 0, h = a constant for Mertens' function M(x) using arithmetic and analytic arguments based on theorems for the prime counting function pi(x) and the series \sum \frac{\mu(k)}{k}. The formula is evaluated using limit theorems to give: an alternative proof of \lim_{x \to \infty} \frac{|M(x)|}{x} = 0, a new disproof of Mertens' conjecture, proof of the Odlyzko--te Riele conjecture and a disproof of the Riemann hypothesis based on Littlewood's equivalence theorem.