Biases in prime factorizations and Liouville functions for arithmetic progressions
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 1-25.

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.

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.

Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1066
Classification: 11A51, 11N13, 11N37, 11F66
Keywords: Liouville function, prime factorization, arithmetic progressions, Pólya’s conjecture
Humphries, Peter 1; Shekatkar, Snehal M. 2; Wong, Tian An 3

1 Department of Mathematics University College London Gower Street London WC1E 6BT, United Kingdom
2 Centre for Modeling and Simulation S.P. Pune University Pune, Maharashtra, 411007 India
3 Smith College 44 College Lane Northampton06013 MA, USA
@article{JTNB_2019__31_1_1_0,
author = {Humphries, Peter and Shekatkar, Snehal M. and Wong, Tian An},
title = {Biases in prime factorizations and {Liouville} functions for arithmetic progressions},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {1--25},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {31},
number = {1},
year = {2019},
doi = {10.5802/jtnb.1066},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jtnb.1066/}
}
TY  - JOUR
AU  - Humphries, Peter
AU  - Shekatkar, Snehal M.
AU  - Wong, Tian An
TI  - Biases in prime factorizations and Liouville functions for arithmetic progressions
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
SP  - 1
EP  - 25
VL  - 31
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - http://archive.numdam.org/articles/10.5802/jtnb.1066/
DO  - 10.5802/jtnb.1066
LA  - en
ID  - JTNB_2019__31_1_1_0
ER  - 
%0 Journal Article
%A Humphries, Peter
%A Shekatkar, Snehal M.
%A Wong, Tian An
%T Biases in prime factorizations and Liouville functions for arithmetic progressions
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 1-25
%V 31
%N 1
%I Société Arithmétique de Bordeaux
%U http://archive.numdam.org/articles/10.5802/jtnb.1066/
%R 10.5802/jtnb.1066
%G en
%F JTNB_2019__31_1_1_0
Humphries, Peter; Shekatkar, Snehal M.; Wong, Tian An. Biases in prime factorizations and Liouville functions for arithmetic progressions. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 1-25. doi : 10.5802/jtnb.1066. http://archive.numdam.org/articles/10.5802/jtnb.1066/

[1] Akbary, Amir; Ng, Nathan; Shahabi, Majid Limiting distributions of the classical error terms of prime number theory, Q. J. Math, Volume 65 (2014) no. 3, pp. 743-780 | DOI | MR

[2] Best, D. G.; Trudgian, Timothy S. Linear relations of zeroes of the zeta-function, Math. Comput., Volume 84 (2015) no. 294, pp. 2047-2058 | DOI | MR | Zbl

[3] Borwein, Peter; Choi, Stephen K. K.; Coons, Michael Completely multiplicative functions taking values in $\left\{-1,1\right\}$, Trans. Am. Math. Soc., Volume 362 (2010) no. 12, pp. 6279-6291 | DOI | MR

[4] Borwein, Peter; Ferguson, Ron; Mossinghoff, Michael J. Sign changes in sums of the Liouville function, Math. Comput., Volume 77 (2008) no. 263, pp. 1681-1694 | DOI | MR

[5] Brent, Richard P.; van de Lune, Jan A note on Pólya’s observation concerning Liouville’s function, Herman J. J. te Riele Liber Amicorum, CWI, 2010, pp. 92-97 (https://arxiv.org/abs/1112.4911)

[6] Changa, M. E. On the sums of multiplicative functions over numbers, all of whose divisors lie in a given arithmetic progression, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 69 (2005) no. 2, pp. 205-220 | DOI | MR | Zbl

[7] Erdős, Pál; Kac, Mark The Gaussian law of errors in the theory of additive number theoretic functions, Am. J. Math., Volume 62 (1940), pp. 738-742 | DOI | MR | Zbl

[8] Haselgrove, Colin B. A disproof of a conjecture of Pólya, Mathematika, Volume 5 (1958), pp. 141-145 | DOI | MR | Zbl

[9] Hughes, C. P.; Keating, Jonathan P.; O’Connell, Neil Random matrix theory and the derivative of the Riemann zeta-function, Proc. R. Soc. Lond., Ser. A, Volume 456 (2000) no. 2003, pp. 2611-2627 | DOI | MR | Zbl

[10] Humphries, Peter The distribution of weighted sums of the Liouville function and Pólya’s conjecture, J. Number Theory, Volume 133 (2013) no. 2, pp. 545-582 | DOI | MR

[11] Ingham, Albert E. On two conjectures in the theory of numbers, Am. J. Math., Volume 64 (1942), pp. 313-319 | DOI | MR | Zbl

[12] Karatsuba, Anatoliĭ A. On a property of the set of prime numbers, Usp. Mat. Nauk, Volume 66 (2011) no. 2, pp. 3-14 | DOI | MR

[13] Landau, Edmund Über die Anzahl der Gitterpunkte in gewissen Bereichen. IV, GÃ¶tt. Nachr., Volume 1924 (1924), pp. 137-150 | Zbl

[14] Languasco, Alessandro; Zaccagnini, Alessandro On the constant in the Mertens product for arithmetic progressions. II. Numerical values, Math. Comput., Volume 78 (2009) no. 265, pp. 315-326 | DOI | MR

[15] Languasco, Alessandro; Zaccagnini, Alessandro On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approximatio, Comment. Math., Volume 42 (2010) no. 1, pp. 17-27 | DOI | MR | Zbl

[16] Matomäki, Kaisa; Radziwiłł, Maksym Multiplicative functions in short intervals, Ann. Math., Volume 183 (2016) no. 3, pp. 1015-1056 | DOI | MR

[17] Meng, Xianchang The distribution of $k$-free numbers and the derivative of the Riemann zeta-function, Math. Proc. Camb. Philos. Soc., Volume 162 (2017) no. 2, pp. 293-317 | DOI | MR

[18] Meng, Xianchang Large bias for integers with prime factors in arithmetic progressions, Mathematika, Volume 64 (2018) no. 1, pp. 237-252 | DOI | MR

[19] Montgomery, Hugh L.; Vaughan, Robert C. Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2007, xviii+552 pages | MR

[20] Pólya, George Verschiedene bemerkungen zur zahlentheorie., Deutsche Math.-Ver., Volume 28 (1919), pp. 31-40 | Zbl

Cited by Sources: