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

Aymone, Marco

Comptes Rendus. Mathématique, Tome 362 (2024) no. G8, 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 ${\sum }_{n\le x}{f}^{*}\left(n\right)$ and ${\sum }_{n\le x}\frac{f\left(n\right)}{\sqrt{n}}$ change sign infinitely often as $x\to \infty$, almost surely. The case ${\sum }_{n\le x}\frac{f^{*}\left(n\right)}{\sqrt{n}}$ 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 ${\sum }_{n\le x}{f}^{*}\left(n\right)$ et ${\sum }_{n\le x}\frac{f\left(n\right)}{\sqrt{n}}$ changent de signe infiniment souvent comme $x\to \infty$, presque sûrement. Le cas ${\sum }_{n\le x}\frac{f^{*}\left(n\right)}{\sqrt{n}}$ reste une question ouverte et nous soulignons la possibilité de seulement un nombre fini de changements de signe, avec probabilité positive.

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

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