Random Chowla's Conjecture for Rademacher Multiplicative Functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918374440697856 |
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| author | Chinis, Jake Shala, Besfort |
| author_facet | Chinis, Jake Shala, Besfort |
| contents | We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb Z[x]$ that is a product of at least two distinct linear factors or an irreducible quadratic satisfying a natural condition, there exists a constant $κ_P>0$ such that
\[
\frac{1}{\sqrt{κ_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1),
\]
as $N\rightarrow\infty$, where convergence is in distribution to a standard (real) Gaussian. This confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu.
We also study large fluctuations of $\sum_{n\leq N}f(n^2+1)$ and show that there almost surely exist arbitrarily large values of $N$ such that \[ \Big|\sum_{n\leq N}f(n^2+1)\Big|\gg \sqrt{N \log\log N}. \] This matches the bound one expects from the law of iterated logarithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05952 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random Chowla's Conjecture for Rademacher Multiplicative Functions Chinis, Jake Shala, Besfort Number Theory Probability 11N37, 11N64, 11D45 We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb Z[x]$ that is a product of at least two distinct linear factors or an irreducible quadratic satisfying a natural condition, there exists a constant $κ_P>0$ such that \[ \frac{1}{\sqrt{κ_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1), \] as $N\rightarrow\infty$, where convergence is in distribution to a standard (real) Gaussian. This confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu. We also study large fluctuations of $\sum_{n\leq N}f(n^2+1)$ and show that there almost surely exist arbitrarily large values of $N$ such that \[ \Big|\sum_{n\leq N}f(n^2+1)\Big|\gg \sqrt{N \log\log N}. \] This matches the bound one expects from the law of iterated logarithm. |
| title | Random Chowla's Conjecture for Rademacher Multiplicative Functions |
| topic | Number Theory Probability 11N37, 11N64, 11D45 |
| url | https://arxiv.org/abs/2409.05952 |