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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.


[1] Amir Akbary, Nathan Ng & Majid Shahabi, Limiting distributions of the classical error terms of prime number theory, Q. J. Math 65 (2014), p. 743-780 Article |  MR 3261965
[2] D. G. Best & Timothy S. Trudgian, Linear relations of zeroes of the zeta-function, Math. Comput. 84 (2015), p. 2047-2058 Article |  MR 3335903
[3] Peter Borwein, Stephen K. K. Choi & Michael Coons, Completely multiplicative functions taking values in $\lbrace -1,1\rbrace $, Trans. Am. Math. Soc. 362 (2010), p. 6279-6291 Article
[4] Peter Borwein, Ron Ferguson & Michael J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comput. 77 (2008), p. 1681-1694 Article
[5] Richard P. Brent & Jan van de Lune, A note on Pólya’s observation concerning Liouville’s function, Herman J. J. te Riele Liber Amicorum, CWI, 2010, p. 92–97
[6] M. E. Changa, On the sums of multiplicative functions over numbers, all of whose divisors lie in a given arithmetic progression, Izv. Ross. Akad. Nauk, Ser. Mat. 69 (2005), p. 205-220 Article
[7] Pál Erdős & Mark Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Am. J. Math. 62 (1940), p. 738-742 Article
[8] Colin B. Haselgrove, A disproof of a conjecture of Pólya, Mathematika 5 (1958), p. 141-145 Article
[9] C. P. Hughes, Jonathan P. Keating & Neil O’Connell, Random matrix theory and the derivative of the Riemann zeta-function, Proc. R. Soc. Lond., Ser. A 456 (2000), p. 2611-2627 Article
[10] Peter Humphries, The distribution of weighted sums of the Liouville function and Pólya’s conjecture, J. Number Theory 133 (2013), p. 545-582 Article
[11] Albert E. Ingham, On two conjectures in the theory of numbers, Am. J. Math. 64 (1942), p. 313-319 Article |  MR 6202
[12] Anatoliĭ A. Karatsuba, On a property of the set of prime numbers, Usp. Mat. Nauk 66 (2011), p. 3-14 Article
[13] Edmund Landau, Über die Anzahl der Gitterpunkte in gewissen Bereichen. IV, Gött. Nachr. 1924 (1924), p. 137-150
[14] Alessandro Languasco & Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II. Numerical values, Math. Comput. 78 (2009), p. 315-326 Article |  MR 2448709
[15] Alessandro Languasco & Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approximatio, Comment. Math. 42 (2010), p. 17-27 Article
[16] Kaisa Matomäki & Maksym Radziwiłł, Multiplicative functions in short intervals, Ann. Math. 183 (2016), p. 1015-1056 Article
[17] Xianchang Meng, The distribution of $k$-free numbers and the derivative of the Riemann zeta-function, Math. Proc. Camb. Philos. Soc. 162 (2017), p. 293-317 Article |  MR 3604916
[18] Xianchang Meng, Large bias for integers with prime factors in arithmetic progressions, Mathematika 64 (2018), p. 237-252 Article |  MR 3778223
[19] Hugh L. Montgomery & Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, 2007  MR 2378655
[20] George Pólya, Verschiedene bemerkungen zur zahlentheorie., Deutsche Math.-Ver. 28 (1919), p. 31-40