The distribution of partial sums of random multiplicative functions with a large prime factor
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912265743106048 |
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| author | Hardy, Seth |
| author_facet | Hardy, Seth |
| contents | For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The distribution of partial sums of random multiplicative functions with a large prime factor Hardy, Seth Number Theory Probability 11K65 For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable. |
| title | The distribution of partial sums of random multiplicative functions with a large prime factor |
| topic | Number Theory Probability 11K65 |
| url | https://arxiv.org/abs/2503.06256 |