Small differences between prime numbers.

Maier, Helmut

doi:10.1307/mmj/1029003814

Cited by 47 publications

(42 citation statements)

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“…This bound was later improved by Huxley [20], [21] to Ä 5=8 C O.1= /, by Goldston and Yıldırım [13] to Ä . p 1=2/ 2 , and by Maier [23] to Ä e . 5=8 C O .1= //.…”

Section: Introductionmentioning

confidence: 95%

Primes in tuples I

2009

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We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is,We will quantify this result further in a later paper.

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“…This bound was later improved by Huxley [20], [21] to Ä 5=8 C O.1= /, by Goldston and Yıldırım [13] to Ä . p 1=2/ 2 , and by Maier [23] to Ä e . 5=8 C O .1= //.…”

Section: Introductionmentioning

confidence: 95%

Primes in tuples I

2009

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“…Other landmark results in the area are the works of Bombieri and Davenport [3], Huxley [20], and Maier [22], who introduced several new ideas to this study and progressively reduced the lim inf to ≤ 0.24 . .…”

Section: Small Gapsmentioning

confidence: 99%

Untitled

2007

Bull. Amer. Math. Soc.

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“…Indeed, Cramer's model suggests that the asymptotic formula (2.2) may hold for extremely short intervals, like y = (log x) A with any constant A > 2. On the other hand H. Maier [54] showed that the asymptotic formula (2.2) fails even for some larger y = y(x). His idea is quite simple, yet the consequences are very surprising (see more observations in the article by J. Friedlander [21]).…”

Section: Gaps Between Primesmentioning

confidence: 99%

“…Of course, for small gaps we expect to have d n = 2 infinitely often (the twin prime conjecture). The problem of finding small gaps between primes sparked a great deal of interest (see Bombieri-Davenport [7], Huxley [42] and Maier [54]). Just a year ago the world was stunned by the following result:…”

Section: Gaps Between Primesmentioning

confidence: 99%

Prime numbers and <var>L</var>-functions

Iwaniec¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk I will show a panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory. Mathematics Subject Classification (2000). Primary L20; Secondary N05.

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Small differences between prime numbers.

Cited by 47 publications

References 0 publications

Primes in tuples I

Primes in tuples I

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Prime numbers and <var>L</var>-functions

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