The distribution of sums and products of additive functions
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 103-131.

The celebrated Erdős–Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening years, similar normal distribution laws have been shown to hold for certain non-additive functions and for amenable arithmetic functions over certain subsets of the natural numbers. Continuing in this vein, we show that if g 1 (n),...,g k (n) is a collection of functions satisfying certain mild hypotheses for which an Erdős–Kac-type normal distribution law holds, and if Q(x 1 ,...,x k ) is a polynomial with nonnegative real coefficients, then Q(g 1 (n),...,g k (n)) also obeys a normal distribution law. We also show that a similar result can be obtained if the set of inputs n is restricted to certain subsets of the natural numbers, such as shifted primes. Our proof uses the method of moments. We conclude by providing examples of our theorem in action.

Le célèbre théorème d’Erdős–Kac dit, en substance, que les valeurs d’une fonction additive satisfaisant certaines hypothèses faibles, sont normalement distribuées. Au cours des dernières décennies, il a été démontré que des lois similaires de distribution normale s’appliquent à certaines fonctions non additives et à des fonctions arithmétiques adaptées à certains sous-ensembles de l’ensemble des nombres naturels. En poursuivant dans cette veine, nous montrons que si g 1 (n),...,g k (n) est un ensemble de fonctions satisfaisant certaines hypothèses légères pour lesquelles une loi de distribution normale de type Erdős–Kac est valide, et si Q(x 1 ,...,x k ) est un polynôme à coefficients non négatifs, alors Q(g 1 (n),...,g k (n)) obéit également à une loi de distribution normale. Nous montrons également qu’un résultat similaire peut être obtenu si l’ensemble des entrées n est limité à certains sous-ensembles de nombres naturels, tels que les nombres premiers décalés. Notre preuve utilise la méthode des moments. Nous concluons en illustrant notre théorème sur quelques exemples.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1113
Classification: 11N60,  11N37
Keywords: loi de distribution, fonction additive, méthode des moments
Greg Martin 1; Lee Troupe 2

1 Department of Mathematics University of British Columbia Room 121, 1984 Mathematics Road Vancouver, BC V6T 1Z2, Canada
2 Department of Mathematics Mercer University 1501 Mercer University Drive Macon, GA 31207, USA
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Greg Martin; Lee Troupe. The distribution of sums and products of  additive functions. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 103-131. doi : 10.5802/jtnb.1113. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1113/

[1] Amir Akbary; Kyle Hambrook A variant of the Bombieri–Vinogradov theorem with explicit constants and applications, Math. Comput., Volume 84 (2015) no. 294, pp. 1901-1932 | Article | MR: 3335897 | Zbl: 1341.11053

[2] Krishnaswami Alladi An Erdős–Kac theorem for integers without large prime factors, Acta Arith., Volume 49 (1987) no. 1, pp. 81-105 | Article | Zbl: 0627.10030

[3] Pál Erdős; Mark Kac The Gaussian law of errors in the theory of additive number theoretic functions, Am. J. Math., Volume 62 (1940) no. 1, pp. 343-352 | MR: 2374 | Zbl: 0024.10203

[4] Andrew Granville; Kannan Soundararajan Sieving and the Erdős–Kac theorem, Equidistribution in number theory, an introduction (NATO Science Series II: Mathematics, Physics and Chemistry) Volume 237, Springer, 2007, pp. 15-27 | Article | Zbl: 1145.11071

[5] Heini Halberstam On the distribution of additive number-theoretic functions, J. Lond. Math. Soc., Volume 30 (1955), pp. 43-53 | Article | MR: 66406 | Zbl: 0064.04202

[6] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, Colloquium Publications, Volume 53, American Mathematical Society, 2004 | MR: 2061214 | Zbl: 1059.11001

[7] Greg Martin; Lee Troupe The distribution of the number of subgroups of the multiplicative group, J. Aust. Math. Soc., Volume 108 (2020) no. 1, pp. 46-97 | Article | MR: 4052936 | Zbl: 07154933

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