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

TL;DR

This paper explores properties of the prime counting function (x) that resemble logarithmic and absolute value functions, revealing new insights and applications related to its behavior.

Contribution

It introduces novel properties of (x) akin to logarithmic and absolute value functions, and examines their applications.

Findings

Identifies properties of (x) similar to logarithm and absolute value functions.

Demonstrates applications of these properties in number theory.

Provides new perspectives on the behavior of (x).

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.