Oscillations of random multiplicative functions under initial bias
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911539226738688 |
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| author | Angelo, Rodrigo Xu, Max Wenqiang |
| author_facet | Angelo, Rodrigo Xu, Max Wenqiang |
| contents | We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-ε}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow \infty$. This solves a conjecture of Kucheriaviy, who has a complementary result showing this exponent is sharp. We also prove that almost surely the partial sums of $\sum\frac{f(n)}{\sqrt{n}}$ change signs infinitely many times, solving a problem of Aymone. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14447 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oscillations of random multiplicative functions under initial bias Angelo, Rodrigo Xu, Max Wenqiang Number Theory Probability We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-ε}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow \infty$. This solves a conjecture of Kucheriaviy, who has a complementary result showing this exponent is sharp. We also prove that almost surely the partial sums of $\sum\frac{f(n)}{\sqrt{n}}$ change signs infinitely many times, solving a problem of Aymone. |
| title | Oscillations of random multiplicative functions under initial bias |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2411.14447 |