Comptes Rendus
Research article - Number theory
Sign changes of the partial sums of a random multiplicative function II
Marco Aymone1
1 Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Av. Antônio Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, Brazil
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 895-901.

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers f, and Rademacher random completely multiplicative functions f * . We prove that the partial sums ∑ n≤x f * (n) and ∑ n≤x f(n) n change sign infinitely often as x→∞, almost surely. The case ∑ n≤x f * (n) n is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.

Nous étudions deux modèles de fonctions multiplicatives aléatoires : les fonctions multiplicatives aléatoires de Rademacher supportées sur les entiers sans carrés f, et les fonctions multiplicatives aléatoires complètement multiplicatives de Rademacher f * . Nous prouvons que les sommes partielles ∑ n≤x f * (n) et ∑ n≤x f(n) n changent de signe infiniment souvent comme x→∞, presque sûrement. Le cas ∑ n≤x f * (n) n reste une question ouverte et nous soulignons la possibilité de seulement un nombre fini de changements de signe, avec probabilité positive.

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Accepted:
Published online:
DOI: 10.5802/crmath.615
Classification: 11K65, 11N99
Keywords: Random multiplicative functions, Oscillation theorems
Mots-clés : Fonctions multiplicatives aléatoires, théorèmes d’oscillation

Marco Aymone 1

1 Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Av. Antônio Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, Brazil
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Marco Aymone. Sign changes of the partial sums of a random multiplicative function II. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 895-901. doi : 10.5802/crmath.615. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.615/

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