Sets of integers satisfying Bateman-Horn statistics
Abstract
In 1962, Bateman and Horn conjectured precise asymptotics for the count of positive integers n x for which f1(n), ..., fk(n) are all prime, where (f1, ..., fk) 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 "f1(n), ..., fk(n) are all prime" with "f1(n), ..., fk(n) all lie in the random set." In particular, sets of integers satisfying Bateman-Horn are plentiful.
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