The Poisson Tail Conjecture for Primes in Short Intervals
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866916037872582656 |
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| author | Jha, Abhishek |
| author_facet | Jha, Abhishek |
| contents | In 1976, Gallagher showed that, conditional on the Hardy--Littlewood conjectures, the number of primes below $x$ in a randomly chosen short interval of length $λ\log x$ asymptotically follows a Poisson distribution with mean $λ$. Correspondingly, the normalized gaps between consecutive primes follow an exponential distribution, provided that the scaling parameter $λ$ is fixed. We investigate the validity and limitations of the associated folklore Poisson Tail Conjecture as $λ$ is allowed to grow. For \edit{slowly growing} $λ$, and conditional on a strong variant of the Hardy--Littlewood conjectures, we establish asymptotics demonstrating that the local counting statistics rigorously align with these predictions. Furthermore, we identify a phase transition and explore the breakdown of these distributions for larger $λ$, capturing the precise deviations when $λ$ grows slower than any fixed power of $\log x$. The proof relies on a novel combination of extremal interval sieve estimates and concentration inequalities from probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23014 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Poisson Tail Conjecture for Primes in Short Intervals Jha, Abhishek Number Theory Probability 11N05, 11B83 (Primary) In 1976, Gallagher showed that, conditional on the Hardy--Littlewood conjectures, the number of primes below $x$ in a randomly chosen short interval of length $λ\log x$ asymptotically follows a Poisson distribution with mean $λ$. Correspondingly, the normalized gaps between consecutive primes follow an exponential distribution, provided that the scaling parameter $λ$ is fixed. We investigate the validity and limitations of the associated folklore Poisson Tail Conjecture as $λ$ is allowed to grow. For \edit{slowly growing} $λ$, and conditional on a strong variant of the Hardy--Littlewood conjectures, we establish asymptotics demonstrating that the local counting statistics rigorously align with these predictions. Furthermore, we identify a phase transition and explore the breakdown of these distributions for larger $λ$, capturing the precise deviations when $λ$ grows slower than any fixed power of $\log x$. The proof relies on a novel combination of extremal interval sieve estimates and concentration inequalities from probability. |
| title | The Poisson Tail Conjecture for Primes in Short Intervals |
| topic | Number Theory Probability 11N05, 11B83 (Primary) |
| url | https://arxiv.org/abs/2605.23014 |