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 e-prints
- Pub Date:
- April 2017
- DOI:
- 10.48550/arXiv.1704.07979
- arXiv:
- arXiv:1704.07979
- Bibcode:
- 2017arXiv170407979H
- Keywords:
-
- Mathematics - Number Theory;
- 11A51;
- 11N13;
- 11N37;
- 11F66
- E-Print:
- 25 pages, 6 figures