# Biases in prime factorizations and Liouville functions for arithmetic progressions

## TL;DR

This paper refines the Liouville function for primes in arithmetic progressions to reveal biases in prime appearances within factorizations, supported by numerical evidence and related to Pólya's conjecture.

## Contribution

It introduces a new refinement of the Liouville function for primes in arithmetic progressions and uncovers biases in prime factorizations not previously documented.

## Key findings

- Primes of the form 4k+1 tend to appear an even number of times more often.
- Biases in prime appearances are observed in the prime factorizations of integers.
- Numerical evidence supports variants of Pólya's conjecture related to these biases.

## Abstract

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of P\'olya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.