Sign changes of the partial sums of a random multiplicative function II

Marco Aymone

doi:10.5802/crmath.615

Article de recherche - Théorie des nombres

Sign changes of the partial sums of a random multiplicative function II
[Changements de signe des sommes partielles d’une fonction multiplicative aléatoire II]

Marco Aymone ¹

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.

Résumé
Abstract

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 $\sum_{n \leq x} f^{*} (n)$ et $\sum_{n \leq x} \frac{f (n)}{\sqrt{n}}$ changent de signe infiniment souvent comme $x \to \infty$ , presque sûrement. Le cas $\sum_{n \leq x} \frac{f^{*} (n)}{\sqrt{n}}$ reste une question ouverte et nous soulignons la possibilité de seulement un nombre fini de changements de signe, avec probabilité positive.

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 $\sum_{n \leq x} f^{*} (n)$ and $\sum_{n \leq x} \frac{f (n)}{\sqrt{n}}$ change sign infinitely often as $x \to \infty$ , almost surely. The case $\sum_{n \leq x} \frac{f^{*} (n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.

Reçu le : 2023-10-19
Accepté le : 2024-01-19
Publié le : 2024-10-07

DOI : 10.5802/crmath.615

Classification : 11K65, 11N99
Keywords: Random multiplicative functions, Oscillation theorems
Mot clés : Fonctions multiplicatives aléatoires, théorèmes d’oscillation

Affiliations des auteurs :

Marco Aymone ¹

¹ Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Av. Antônio Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, Brazil

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Sign changes of the partial sums of a random multiplicative function {II}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {895--901},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.615},
     language = {en},
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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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