Logarithmic and absolute-value like properties of $\pi (x)$ with applications

Abstract

We derive properties of $\pi (x)$ reminiscent of those of the logarithm and absolute value functions. Two of these properties are similar to the relations defining the linearity of a function. Several applications of these properties of $\pi (x)$ are also examined.

11 pages. Withdrawn by the author due to a problem with Theorem 3.1, thus also invalidating all results that were based on it