Biases in the distribution of primes in short intervals
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.
Related
- The Poisson Tail Conjecture for primes in short intervals
- Primes in short intervals assuming the Riemann hypothesis
- Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)k
- Bounded intervals containing a given number of primes
- Quasimodular Forms, Partitions, and Primes in Intervals
BibTeX
@misc{freiberg2026biases,
title = {{Biases in the distribution of primes in short intervals}},
author = {Tristan Freiberg},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33692},
eprint = {2609.33692},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33692},
}
RIS
TY - UNPB
TI - Biases in the distribution of primes in short intervals
AU - Tristan Freiberg
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33692
UR - https://arxiv.org/abs/2609.33692
AB - 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.
ER -