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Peter Humphries; Snehal M. Shekatkar; Tian An Wong
Biases in prime factorizations and Liouville functions for arithmetic progressions
Journal de théorie des nombres de Bordeaux, 31 no. 1 (2019), p. 1-25
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Class. Math.: 11A51, 11N13, 11N37, 11F66
Keywords: Liouville function, prime factorization, arithmetic progressions, Pólya’s conjecture

Résumé - Abstract

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.

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