Small gaps between primes exist
Daniel A. Goldston,Yoichi Motohashi,János Pintz,C. Y. Yildirim +3 more
- 01 Apr 2006
- Vol. 82, Iss: 4, pp 61-65
TL;DR: Goldston, Pintz, and Yoldorom as mentioned in this paper showed that an improvement of the Bombieri-Vinogradov prime number theorem would give rise infinitely often to bounded differences between consecutive primes.
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Abstract: 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.
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Citations
Primes in tuples I
TL;DR: In this article, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
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The dichotomy between structure and randomness, arithmetic progressions, and the primes
Terence Tao
- 01 Jan 2006
TL;DR: In this paper, a survey of various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory is presented, and the underlying themes in these arguments are remarkably similar even though the contexts are radically different.
Primes in tuples II
TL;DR: In this paper, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
What is good mathematics
TL;DR: Good mathematical problem-solving (e.g., a major breakthrough on an important mathematical problem); good mathematical technique (i.e., a masterful use of existing methods, or the development of new tools); (ii) good mathematical theory (i, e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results); (iii) Good mathematical insight (e,g. the revelation of an unexpected and intriguing new mathematical phenomenon, connection, or counterexample); (iv) Good Mathematical application (e
•Posted Content
The dichotomy between structure and randomness, arithmetic progressions, and the primes
TL;DR: A survey of various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory can be found in this paper, where the underlying themes in these arguments are remarkably similar even though the contexts are radically different.
71
References
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TL;DR: In this article, it was shown that the number of primes in an interval (n, n + h), averaged over n ≤ N, tends to the limit λ, when n and h tend to infinity in such a way that h ∼ λ log N, with λ a positive constant.
•Book
Le grand crible dans la théorie analytique des nombres
Enrico Bombieri
- 01 Jan 1974
TL;DR: In this article, the conditions générales d'utilisation (http://www.emath.numdam.org/conditions) are defined, i.e., toute utilisation commerciale ou impression systématique is constitutive of an infraction pénale.