Sets of integers satisfying Bateman-Horn statistics
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910184953085952 |
|---|---|
| author | Banks, William Ford, Kevin |
| author_facet | Banks, William Ford, Kevin |
| contents | In 1962, Bateman and Horn conjectured precise asymptotics for the count of positive integers n \le x for which f_1(n), ..., f_k(n) are all prime, where (f_1, ..., f_k) is an admissible k-tuple of polynomials in one variable. We prove that certain random sets of integers almost surely satisfy the Bateman-Horn asymptotics in full generality and with a strong error term, where we have replaced "f_1(n), ..., f_k(n) are all prime" with "f_1(n), ..., f_k(n) all lie in the random set." In particular, sets of integers satisfying Bateman-Horn are plentiful. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01155 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sets of integers satisfying Bateman-Horn statistics Banks, William Ford, Kevin Number Theory 2020: Primary 11N32, Secondary 11N05, 11K65, 60F20 In 1962, Bateman and Horn conjectured precise asymptotics for the count of positive integers n \le x for which f_1(n), ..., f_k(n) are all prime, where (f_1, ..., f_k) is an admissible k-tuple of polynomials in one variable. We prove that certain random sets of integers almost surely satisfy the Bateman-Horn asymptotics in full generality and with a strong error term, where we have replaced "f_1(n), ..., f_k(n) are all prime" with "f_1(n), ..., f_k(n) all lie in the random set." In particular, sets of integers satisfying Bateman-Horn are plentiful. |
| title | Sets of integers satisfying Bateman-Horn statistics |
| topic | Number Theory 2020: Primary 11N32, Secondary 11N05, 11K65, 60F20 |
| url | https://arxiv.org/abs/2605.01155 |