Central limit theorems for random multiplicative functions over function fields
Fuente:
arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866909947158069248 |
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| author | Hoban, Declan Shah, Jibran Iqbal Ismail, Nadya-Catherine Verreault, William Zaman, Asif |
| author_facet | Hoban, Declan Shah, Jibran Iqbal Ismail, Nadya-Catherine Verreault, William Zaman, Asif |
| contents | We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends recent work of Soundararajan and Xu to the function field setting. We apply this characterization to deduce central limit theorems in four cases: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. In doing so, we also establish an explicit Hildebrand inequality for smooth polynomials in short intervals, a function field form of Shiu's theorem for multiplicative functions, and an explicit Chebyshev bound for rough polynomials in short intervals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22905 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Central limit theorems for random multiplicative functions over function fields Hoban, Declan Shah, Jibran Iqbal Ismail, Nadya-Catherine Verreault, William Zaman, Asif Number Theory Probability We provide a sufficient characterization for subsets $\mathcal{A}$ of the polynomial ring $\mathbb{F}_q[t]$ for which partial sums of Steinhaus random multiplicative functions approach a complex standard normal distribution. This extends recent work of Soundararajan and Xu to the function field setting. We apply this characterization to deduce central limit theorems in four cases: polynomials in short intervals, polynomials with few prime factors, shifted primes, and rough polynomials. In doing so, we also establish an explicit Hildebrand inequality for smooth polynomials in short intervals, a function field form of Shiu's theorem for multiplicative functions, and an explicit Chebyshev bound for rough polynomials in short intervals. |
| title | Central limit theorems for random multiplicative functions over function fields |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2511.22905 |