Title: Unexpected biases in prime factorizations and Liouville functions for arithmetic progressions,
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
Pólya's conjecture, supported by extensive numerical evidence, and its
relation to other conjectures.