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.

25 pages, 6 figures

References in corpus (3)

• Unexpected biases in the distribution of consecutive primes
• Large bias for integers with prime factors in arithmetic progressions
• Bounding sums of the Möbius function over arithmetic progressions

Cited by in corpus (1)

• An annotated bibliography for comparative prime number theory