Sign changes of the partial sums of a random multiplicative function II
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910307098558464 |
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| author | Aymone, Marco |
| author_facet | Aymone, Marco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14682 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sign changes of the partial sums of a random multiplicative function II Aymone, Marco Number Theory Probability 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. |
| title | Sign changes of the partial sums of a random multiplicative function II |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2303.14682 |