# On the prime counting function and the partial sum of reciprocals of odd primes

## TL;DR

This paper introduces a new function that tests primality, factorizes composites, and expresses the prime counting function using sums over primes and a variant of the prime factor count.

## Contribution

It provides a novel approach to relate the prime counting function to sums over primes and a new prime factorization-related function.

## Key findings

- Derived a closed-form expression for π(n^2)
- Developed a primality testing and factorization function
- Connected prime sums with the prime counting function

## Abstract

We present a function that tests for primality, factorizes composites and builds a closed form expression of $\pi(n^2)$ in terms of $\sum_{3 \leq p \leq n} \frac{1}{p}$ and a weaker version of $\omega(n)$.