A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields

Kaptan, Deniz Ali

doi:10.1007/s10474-013-0296-x

Cited by 5 publications

(4 citation statements)

References 6 publications

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“…This fact and by implication Lemma 3 seem to be well known (see e.g. [10] or [3]), but we indicate the proof below.…”

Section: 2mentioning

confidence: 61%

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Sums of singular series and primes in short intervals in algebraic number fields

Kuperberg¹,

Rodgers²,

Roditty-Gershon³

2020

Preprint

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Gross and Smith have put forward generalizations of Hardy -Littlewood twin prime conjectures for algebraic number fields. We estimate the behavior of sums of a singular series that arises in these conjectures, up to lower order terms. More exactly, where S(η) is the singular series, we find asymptotic formulas for smoothed sums of S(η) − 1.Based upon Gross and Smith's conjectures, we use our result to suggest that for large enough 'short intervals' in an algebraic number field K, the variance of counts of prime elements in a random short interval deviates from a Cramér model prediction by a universal factor, independent of K. The conjecture over number fields generalizes a classical conjecture of Goldston and Montgomery over the integers. Numerical data is provided supporting the conjecture.

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“…This fact and by implication Lemma 3 seem to be well known (see e.g. [10] or [3]), but we indicate the proof below.…”

Section: 2mentioning

confidence: 61%

“…), and so using Proposition 1 and the subsequent evaluation (10) to keep track of E K (X; H) asymptotically, the above simplifies to…”

Section: Primes In Short Intervals: a Heuristic Argumentmentioning

confidence: 99%

Sums of singular series and primes in short intervals in algebraic number fields

Kuperberg¹,

Rodgers²,

Roditty-Gershon³

2020

Preprint

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“…There is a major misprint in the statement of the main result of my paper [1] in all three instances where it was stated, namely, a logarithm is missing from the denominator. The second displayed equation of the abstract, the second displayed equation on page 85, and the final displayed equation of the paper in the theorem statement on page 111 should all read lim inf…”

mentioning

confidence: 94%

Erratum to: “A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields”

Kaptan

2014

Acta Math. Hungar.

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“…In the spirit of [1,4,5,8] it is natural to consider gaps between products of two primes in number fields. Before stating our main result of this article, we will fix some notations.…”

Section: Introductionmentioning

confidence: 99%

Bounded gaps between product of two primes in imaginary quadratic number fields

2022

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We study the gaps between products of two primes in imaginary quadratic number fields using a combination of the methods of Goldston–Graham–Pintz–Yildirim (Proc Lond Math Soc 98:741–774, 2009), and Maynard (Ann Math 181:383–413, 2015). An important consequence of our main theorem is existence of infinitely many pairs $$\alpha _1, \alpha _2$$ α 1 , α 2 which are product of two primes in the imaginary quadratic field K such that $$|\sigma (\alpha _1-\alpha _2)|\le 2$$ | σ ( α 1 - α 2 ) | ≤ 2 for all embeddings $$\sigma $$ σ of K if the class number of K is one and $$|\sigma (\alpha _1-\alpha _2)|\le 8$$ | σ ( α 1 - α 2 ) | ≤ 8 for all embeddings $$\sigma $$ σ of K if the class number of K is two.

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A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields

Cited by 5 publications

References 6 publications

Sums of singular series and primes in short intervals in algebraic number fields

Sums of singular series and primes in short intervals in algebraic number fields

Erratum to: “A generalization of the Goldston–Pintz–Yildirim prime gaps result to number fields”

Bounded gaps between product of two primes in imaginary quadratic number fields

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