Journal ArticleDOI
Bounded gaps between primes
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Abstract:
It is proved that
lim inf n?8 (p n+1 -p n )<7×10 7 , where p n is the n -th prime.
Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purposeread more
Citations
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Journal ArticleDOI
Small gaps between primes
TL;DR: In this paper, the GPY sieve method for studying prime k-tuples and small gaps between primes was introduced and it was shown that for each k, the prime k -tuples conjecture holds for a positive proportion of admissible k-toples.
Journal ArticleDOI
Variants of the Selberg sieve, and bounded intervals containing many primes
TL;DR: In particular, this paper showed that for any admissible triple (h1,h2,h3), there are infinitely many n for which at least two of n+h 1,n+h 2,h 3 are prime, and also showed that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both.
Journal ArticleDOI
Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals
TL;DR: In the companion paper (Part II), it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto Ramanujan subspaces.
Book ChapterDOI
Primes in Arithmetic Progressions
TL;DR: In this paper, it is shown that Dirichlet's original argument also was a real-variable argument, since by that time, a hundred years later, complex analysis was well-established.
Journal ArticleDOI
New equidistribution estimates of Zhang type
Wouter Castryck,Étienne Fouvry,Gergely Harcos,Emmanuel Kowalski,Philippe Michel,Paul D. Nelson,E. Paldi,János Pintz,Andrew V. Sutherland,Terence Tao,Xiao-Feng Xie +10 more
TL;DR: For arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese remainder theorem conditions, the authors obtained an exponent of distribution 1/2 + 7/300.
References
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Journal ArticleDOI
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TL;DR: In this article, the authors implique l'accord avec les conditions generales d'utilisation (http://www.numdam.org/legal.php).
Journal ArticleDOI
Numbers of solutions of equations in finite fields
TL;DR: In this paper, it was shown that for a prime of the form p = 4n + 1, the number of solutions of any congruence ax − by ≡ 1 (mod p) for a given biquadratic character of 2 mod p can be computed using Gaussian sums of order 3.
Journal ArticleDOI
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.
Journal ArticleDOI
A Brun-Titschmarsh theorem for multiplicative functions.
TL;DR: In this paper, the problem of estimating the number of positive integers not exceeding χ having no prime factor greater than y is reduced to estimating the Φ(χ,γ) of positive integer not exceeding Φ having no constant in a and k, provided that k
Journal ArticleDOI
On the large sieve
TL;DR: In this paper, a new and improved version of Linnik's large sieve, with some applications, is given, which is based on the Hardy-Littlewood method, and in its most general form it may be considered as an inequality which relates a singular series arising from an integral where S(α) is any exponential sum.
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