Sign changes of the partial sums of a random multiplicative function II
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 $\sum _{n\le x}f^*(n)$ and $\sum _{n\le x}\frac{f(n)}{\sqrt{n}}...
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| Language: | English |
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Académie des sciences
2024-10-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.615/ |
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| author | Aymone, Marco |
| author_facet | Aymone, Marco |
| author_sort | Aymone, Marco |
| collection | DOAJ |
| description | 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 $\sum _{n\le x}f^*(n)$ and $\sum _{n\le x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\rightarrow \infty $, almost surely. The case $\sum _{n\le x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability. |
| format | Article |
| id | doaj-art-4acf36015ba643ab9399e17e85d44d49 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-4acf36015ba643ab9399e17e85d44d492025-02-07T11:22:49ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-10-01362G889590110.5802/crmath.61510.5802/crmath.615Sign changes of the partial sums of a random multiplicative function IIAymone, Marco0Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Av. Antônio Carlos, 6627, CEP 31270-901, Belo Horizonte, MG, BrazilWe 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 $\sum _{n\le x}f^*(n)$ and $\sum _{n\le x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\rightarrow \infty $, almost surely. The case $\sum _{n\le x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.615/Random multiplicative functionsOscillation theorems |
| spellingShingle | Aymone, Marco Sign changes of the partial sums of a random multiplicative function II Comptes Rendus. Mathématique Random multiplicative functions Oscillation theorems |
| title | Sign changes of the partial sums of a random multiplicative function II |
| title_full | Sign changes of the partial sums of a random multiplicative function II |
| title_fullStr | Sign changes of the partial sums of a random multiplicative function II |
| title_full_unstemmed | Sign changes of the partial sums of a random multiplicative function II |
| title_short | Sign changes of the partial sums of a random multiplicative function II |
| title_sort | sign changes of the partial sums of a random multiplicative function ii |
| topic | Random multiplicative functions Oscillation theorems |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.615/ |
| work_keys_str_mv | AT aymonemarco signchangesofthepartialsumsofarandommultiplicativefunctionii |