Biases in the distribution of primes in short intervals

Tristan Freiberg

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arXiv · 2609.33692

Biases in the distribution of primes in short intervals

Tristan Freiberg

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Abstract

Assuming a suitably uniform Hardy--Littlewood prime tuples hypothesis, we obtain a second-order asymptotic for the proportion of short intervals containing a prescribed number of primes, when the interval length is comparable to the average prime spacing. The leading term is Poisson, but the arithmetic correction differs from the binomial correction in Cramér's independent model and predicts a stronger bias toward counts near the mean. The proof combines inclusion--exclusion with singular-series estimates of Montgomery and Soundararajan and a finite-sieve argument that controls the required alternating sums as the truncation order grows. We also give a refined random model that reproduces the correction and numerical comparisons that support the prediction.

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Tristan Freiberg. 2026-09-27. Biases in the distribution of primes in short intervals. https://arxiv.org/abs/2609.33692

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