The Distribution of Prime Numbers

Koukoulopoulos, Dimitris

doi:10.1090/gsm/203

Cited by 39 publications

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“…We then set a = The sum over a equals q p (1 − q −1+εp ) −1 with q denoting a prime number. Since q εp = 1 + O(log q/ log p) for q p, Mertens' estimates [8,Theorem 3.4] imply that the sum over a is log p. We conclude that…”

Section: Preliminariesmentioning

confidence: 68%

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A note on the natural density of product sets

Bettin

Koukoulopoulos

Sanna

2021

Bull. London Math. Soc.

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Section: Preliminariesmentioning

confidence: 68%

“…from the inclusion-exclusion principle that has 2 #P 2 log x = o(x) steps (for example, see [8,Theorem 2.1]). Since p∈P\P (1 − 1/p) ∼ 1 by our assumption that p∈P 1/p < ∞, the proof is complete.…”

Section: Preliminariesmentioning

confidence: 99%

A note on the natural density of product sets

Bettin

Koukoulopoulos

Sanna

2021

Bull. London Math. Soc.

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“…for q large enough by Mertens' estimate [19,Theorem 3.4]. Already the above inequalities show that only the primes (log q) 0.01 can affect the size of ϕ(q)/q.…”

Section: Progress Towards the Duffin-schaeffer Conjecturementioning

confidence: 93%

“…by Mertens' theorem [19,Theorem 3.4]. This is much smaller than e j /j 2 , so (2.10) should fail rarely as j → ∞.…”

Section: Progress Towards the Duffin-schaeffer Conjecturementioning

confidence: 97%

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Rational approximations of irrational numbers

Koukoulopoulos¹

2021

Preprint

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Given quantities ∆ 1 , ∆ 2 , • • • 0, a fundamental problem in Diophantine approximation is to understand which irrational numbers x have infinitely many reduced rational approximations a/q such that |x − a/q| < ∆ q . Depending on the choice of ∆ q and of x, this question may be very hard. However, Duffin and Schaeffer conjectured in 1941 that if we assume a "metric" point of view, the question is governed by a simple zero-one law: writing ϕ for Euler's totient function, we either have ∞ q=1 ϕ(q)∆ q = ∞ and then almost all irrational numbers (in the Lebesgue sense) are approximable, or ∞ q=1 ϕ(q)∆ q < ∞ and almost no irrationals are approximable. We present the history of the Duffin-Schaeffer conjecture and the main ideas behind the recent work of Koukoulopoulos-Maynard that settled it.

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Jordan constants of quaternion algebras over number fields and simple abelian surfaces over fields of positive characteristic

Hwang

2022

Mathematische Nachrichten

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We compute and provide a detailed description on the Jordan constants of the multiplicative subgroup of quaternion algebras over number fields of small degree. As an application, we determine the Jordan constants of the multiplicative subgroup of the endomorphism algebras of simple abelian surfaces over fields of positive characteristic. K E Y W O R D Sabelian surfaces over fields of positive characteristic, Jordan groups and Jordan constants, quaternion algebras over number fields M S C ( 2 0 2 0 )

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The Distribution of Prime Numbers

Cited by 39 publications

References 0 publications

A note on the natural density of product sets

A note on the natural density of product sets

Rational approximations of irrational numbers

Jordan constants of quaternion algebras over number fields and simple abelian surfaces over fields of positive characteristic

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