Positive proportion of small gaps between consecutive primes

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D. A. Goldston, J. Pintz and C. Y. Yıldırım, Positive Proportion of Small Gaps Between Consecutive Primes. The selected text is arXiv:1103.3986v1, posted 21 March 2011. The source PDF is available from arXiv.

The arXiv record, read 2026-10-15, specifies arXiv's non-exclusive distribution license. This does not grant redistribution under an open license. No file of this source is held.

Scope

Write $p_n$ for the $n$th prime and $\pi(x)$ for the prime-counting function. Theorem 1 states that, for each fixed $\eta>0$,

$$\frac{\#\{p_n\leq x:p_{n+1}-p_n\leq\eta\log p_n\}}{\pi(x)} \gg_\eta 1$$

as $x\to\infty$. The fixed-$\eta$ condition matters: the theorem does not assert a positive lower proportion uniform as $\eta\to0$. The introduction separately labels stronger distribution-dependent results and an Elliott–Halberstam-dependent result for triples.

Read status: claims checked. The introduction, the definition of its counting function and Theorems 1–4 were read in the arXiv TeX source at statement depth. Their proofs have not been checked here.

Bears on. Problem 251: this is a comparison for assumptions about short consecutive-prime gaps. The pair-counting result alone does not specify long residue patterns or prove the irrationality of the prime-weighted binary series.