A Characteristic Function for the Primes

TL;DR

The paper introduces a characteristic function that identifies prime numbers and explores its applications in expressing the prime counting function and counting distinct prime factors.

Contribution

It presents a novel characteristic function for primes and demonstrates its use in deriving new formulas for prime counting and prime factorization.

Findings

New expression for the prime counting function

Expression for the number of distinct prime factors

Potential for improved prime number analysis

Abstract

I develop a function that, for any integer $n \geq 2$, takes a value of 1 if $n$ is prime, 0 if $n$ is composite. I also discuss two applications: First, the characteristic function provides a new expression for the prime counting function. Second, the components of the characteristic function point to a new expression that gives the number of distinct prime factors for any integer greater than one.