Certain random sets of integers almost surely obey the full Bateman-Horn asymptotics for polynomial tuples.
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2026 2verdicts
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Conditional on a strong form of the Hardy-Littlewood conjectures, the Poisson tail conjecture for primes in short intervals holds for slowly growing λ with a phase transition identified for larger λ.
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Sets of integers satisfying Bateman-Horn statistics
Certain random sets of integers almost surely obey the full Bateman-Horn asymptotics for polynomial tuples.
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The Poisson Tail Conjecture for Primes in Short Intervals
Conditional on a strong form of the Hardy-Littlewood conjectures, the Poisson tail conjecture for primes in short intervals holds for slowly growing λ with a phase transition identified for larger λ.