Large fluctuations of sums of a random multiplicative function
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910038702948352 |
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| author | Shala, Besfort |
| author_facet | Shala, Besfort |
| contents | Let $f$ be a Rademacher or Steinhaus random multiplicative function. For various arithmetically interesting subsets $\mathcal A\subseteq [1, N]\cap\mathbb N$ such that the distribution of $\sum_{n\in \mathcal A} f(n)$ is approximately Gaussian, we develop a general framework to understand the large fluctuations of the sum. This extends the general central limit theorem framework of Soundararajan and Xu.
In the case when $\mathcal A = (N-H, N]$ is a short interval with admissible $H=H(N)$, we show that almost surely \begin{equation*}
\limsup_{N\to\infty} \frac{\big\lvert\sum_{N-H<n\leq N} f(n)\big\rvert}{\sqrt{H\log \frac{N}H{}}}>0. \end{equation*} When $\mathcal A$ is the set of values of an admissible polynomial $P\in\mathbb Z[x]$, we extend work of Klurman, Shkredov, and Xu, as well as Chinis and the author, showing that almost surely \begin{equation*}
\limsup_{N\to\infty} \frac{\big\lvert\sum_{n\leq N} f(P(n))\big\rvert}{\sqrt{N \log\log N}}>0, \end{equation*} even when $P$ is a product of linear factors over $\mathbb Q$. In this case, we also establish the corresponding almost sure upper bound, matching the law of iterated logarithm.
An important ingredient in our work is bounding the Kantorovich--Wasserstein distance by means of a quantitative martingale central limit theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20086 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large fluctuations of sums of a random multiplicative function Shala, Besfort Number Theory Probability 11K65, 11N64, 60F10, 60G42 Let $f$ be a Rademacher or Steinhaus random multiplicative function. For various arithmetically interesting subsets $\mathcal A\subseteq [1, N]\cap\mathbb N$ such that the distribution of $\sum_{n\in \mathcal A} f(n)$ is approximately Gaussian, we develop a general framework to understand the large fluctuations of the sum. This extends the general central limit theorem framework of Soundararajan and Xu. In the case when $\mathcal A = (N-H, N]$ is a short interval with admissible $H=H(N)$, we show that almost surely \begin{equation*} \limsup_{N\to\infty} \frac{\big\lvert\sum_{N-H<n\leq N} f(n)\big\rvert}{\sqrt{H\log \frac{N}H{}}}>0. \end{equation*} When $\mathcal A$ is the set of values of an admissible polynomial $P\in\mathbb Z[x]$, we extend work of Klurman, Shkredov, and Xu, as well as Chinis and the author, showing that almost surely \begin{equation*} \limsup_{N\to\infty} \frac{\big\lvert\sum_{n\leq N} f(P(n))\big\rvert}{\sqrt{N \log\log N}}>0, \end{equation*} even when $P$ is a product of linear factors over $\mathbb Q$. In this case, we also establish the corresponding almost sure upper bound, matching the law of iterated logarithm. An important ingredient in our work is bounding the Kantorovich--Wasserstein distance by means of a quantitative martingale central limit theorem. |
| title | Large fluctuations of sums of a random multiplicative function |
| topic | Number Theory Probability 11K65, 11N64, 60F10, 60G42 |
| url | https://arxiv.org/abs/2602.20086 |