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Small Gaps Between Primes I
D. A. Goldston,C. Y. Yildirim +1 more
TL;DR: The authors used short divisor sums to approximate prime tuples and moments for primes in short intervals and showed that a positive proportion of consecutive primes are within a quarter of the average spacing between primes.
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Abstract: We use short divisor sums to approximate prime tuples and moments for primes in short intervals By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a quarter of the average spacing between primes
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Citations
The primes contain arbitrarily long arithmetic progressions
Benjamin W. Green,Terence Tao +1 more
TL;DR: In this paper, it was shown that there are arbitrarily long arithmetic progressions of primes and that a large fraction of the primes can be placed inside a pseudorandom set of almost primes with positive relative density.
1K
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.
Restriction theory of the Selberg sieve, with applications
Ben Green,Terence Tao +1 more
TL;DR: In this article, Chen et al. demontrons un theoreme de restriction L 2 -L P for the majorants of suites arithmiques with k-uplets premiers.
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
•Book
The Theory of the Riemann Zeta-Function
E. C. Titchmarsh,D. R. Heath-Brown +1 more
- 05 Feb 1987
TL;DR: The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it one of the most important tools in the study of prime numbers as mentioned in this paper.
4.2K
Primes in arithmetic progressions to large moduli
TL;DR: In this article, the authors present a set of notations for the use of the word "cascade" in the form of a sequence of n-grams, where each node corresponds to a node in a tree.
On the distribution of primes in short intervals
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.
The difference of consecutive primes
TL;DR: Erdös and Turân as mentioned in this paper showed that the primes satisfying (4) also satisfy the first inequality of (3) i΀ = e(ci) is chosen small enough.
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