Biases in prime factorizations and Liouville functions for arithmetic progressions

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 Pólya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.

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