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.
 Publication:

arXiv eprints
 Pub Date:
 April 2017
 DOI:
 10.48550/arXiv.1704.07979
 arXiv:
 arXiv:1704.07979
 Bibcode:
 2017arXiv170407979H
 Keywords:

 Mathematics  Number Theory;
 11A51;
 11N13;
 11N37;
 11F66
 EPrint:
 25 pages, 6 figures