Small gaps between primes exist

By D. Goldston, Yoichi Motohashi, Janos Pintz, C. Yildirim. 2005

55 citations recorded by the source corpus.

In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in (3). While (3) also includes quantitative versions of (0), we are concerned here solely with proving the qualitative (0), which still exhibits all the essentials of the method. We also show here that an improvement of the Bombieri-Vinogradov prime number theorem would give rise infinitely often to bounded differences between consecutive primes. We include a short expository last section. Detailed discussions of quantitative results and a historical review will appear in the publication version of (3) and its continuations.

Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/ff55eb2eba0b9c4b4e9a8a05c44284f60acd163f

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In Algebra and Number Theory

In Geometry and Topology

In Theoretical Computer Science

Citations in the corpus

The dichotomy between structure and randomness, arithmetic progressions, and the primes (2005, background): “7A related enveloping sieve was also used in the recent establishment of narrow gaps in the primes [18].”
Primes in tuples I (2005, methodology): “A more algebraic discussion of d may also be found in [14], [15].”
Primes in tuples II (2007, methodology): “A more algebraic discussion of d may also be found in [14], [15].”