Biases in prime factorizations and Liouville functions for arithmetic progressions

Humphries, Peter; Shekatkar, Snehal M.; Wong, Tian An

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 1-25.

Nous introduisons un raffinement de la fonction classique de Liouville pour les nombres premiers en progressions arithmétiques. En utilisant ces fonctions, nous montrons que l’apparition de nombres premiers dans la factorisation des entiers dépend de la progression arithmétique à laquelle ces nombres premiers appartiennent. Encouragés par des explorations numériques, nous sommes amenés à considérer des analogues de la conjecture de Pólya et à prouver des résultats liés aux changements de signe des fonctions de sommation associées.

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we show that the occurrence of primes in the prime factorizations of integers depends on the arithmetic progressions to which the given primes belong. Supported by numerical tests, we are led to consider analogues of Pólya’s conjecture, and prove results related to the sign changes of the associated summatory functions.

Classification : 11A51, 11N13, 11N37, 11F66
Mots clés : Liouville function, prime factorization, arithmetic progressions, Pólya’s conjecture

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