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arXiv · 2605.01155

Sets of integers satisfying Bateman-Horn statistics

William Banks·Kevin Ford
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Abstract

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.

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William Banks, Kevin Ford. 2026-05-01. Sets of integers satisfying Bateman-Horn statistics. https://arxiv.org/abs/2605.01155

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