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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.11884 |
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| _version_ | 1866908948345389056 |
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| author | Bharadwaj, Abhishek Rodgers, Brad |
| author_facet | Bharadwaj, Abhishek Rodgers, Brad |
| contents | We study the distribution of large prime factors of a random element $u$ of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the large prime factors of $u$ tend to a Poisson-Dirichlet process, while if the sequence has any positive level of distribution the correlation functions of large prime factors tend to a Poisson-Dirichlet process against test functions of restricted support. For sequences with positive level of distribution, we also estimate the probability the largest prime factor of $u$ is greater than $u^{1-ε}$, showing that this probability is $O(ε)$.
Examples of sequences described include shifted primes and values of single-variable irreducible polynomials.
The proofs involve (i) a characterization of the Poisson-Dirichlet process due to Arratia-Kochman-Miller and (ii) an upper bound sieve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11884 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large prime factors of well-distributed sequences Bharadwaj, Abhishek Rodgers, Brad Number Theory We study the distribution of large prime factors of a random element $u$ of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the large prime factors of $u$ tend to a Poisson-Dirichlet process, while if the sequence has any positive level of distribution the correlation functions of large prime factors tend to a Poisson-Dirichlet process against test functions of restricted support. For sequences with positive level of distribution, we also estimate the probability the largest prime factor of $u$ is greater than $u^{1-ε}$, showing that this probability is $O(ε)$. Examples of sequences described include shifted primes and values of single-variable irreducible polynomials. The proofs involve (i) a characterization of the Poisson-Dirichlet process due to Arratia-Kochman-Miller and (ii) an upper bound sieve. |
| title | Large prime factors of well-distributed sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.11884 |