On Hölder maps and prime gaps

Haipeng Chen; Jonathan Fraser

doi:10.14321/realanalexch.46.2.0523

On Hölder maps and prime gaps

Research output: Contribution to journal › Article › peer-review

Abstract

Let p_n denote the nth prime, and consider the function 1/n → 1/p_n which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parametrised family of Cramér type estimates on the gaps between successive primes. Here the parametrisation comes from the Hölder exponent. In particular, we show that Cramér’s conjecture is equivalent to the map 1/n → 1/p_n being Lipschitz. On the other hand, we show that the inverse map 1/p_n → 1/n is Hölder of all orders but not Lipschitz and this is independent of Cramér’s conjecture.

Original language	English
Pages (from-to)	523-532
Journal	Real Analysis Exchange
Volume	46
Issue number	2
DOIs	https://doi.org/10.14321/realanalexch.46.2.0523
Publication status	Published - 8 Nov 2021

Keywords

Prime
Prime gaps
Cramér’s conjecture
Hölder maps
Lipschitz maps

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10.14321/realanalexch.46.2.0523

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Copyright © 2021 Michigan State University Press. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.14321/realanalexch.46.2.0523.
Accepted author manuscript, 279 KB

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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title = "On H{\"o}lder maps and prime gaps",

abstract = "Let pn denote the nth prime, and consider the function 1/n → 1/pn which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that H{\"o}lder continuity of this function is equivalent to a parametrised family of Cram{\'e}r type estimates on the gaps between successive primes. Here the parametrisation comes from the H{\"o}lder exponent. In particular, we show that Cram{\'e}r{\textquoteright}s conjecture is equivalent to the map 1/n → 1/pn being Lipschitz. On the other hand, we show that the inverse map 1/pn → 1/n is H{\"o}lder of all orders but not Lipschitz and this is independent of Cram{\'e}r{\textquoteright}s conjecture.",

keywords = "Prime, Prime gaps, Cram{\'e}r{\textquoteright}s conjecture, H{\"o}lder maps, Lipschitz maps",

author = "Haipeng Chen and Jonathan Fraser",

note = "Funding: The research of H. Chen was funded by China Scholarship Council (File No. 201906150102). J. M.Fraser was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034).",

year = "2021",

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doi = "10.14321/realanalexch.46.2.0523",

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T1 - On Hölder maps and prime gaps

AU - Chen, Haipeng

AU - Fraser, Jonathan

N1 - Funding: The research of H. Chen was funded by China Scholarship Council (File No. 201906150102). J. M.Fraser was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034).

PY - 2021/11/8

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N2 - Let pn denote the nth prime, and consider the function 1/n → 1/pn which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parametrised family of Cramér type estimates on the gaps between successive primes. Here the parametrisation comes from the Hölder exponent. In particular, we show that Cramér’s conjecture is equivalent to the map 1/n → 1/pn being Lipschitz. On the other hand, we show that the inverse map 1/pn → 1/n is Hölder of all orders but not Lipschitz and this is independent of Cramér’s conjecture.

AB - Let pn denote the nth prime, and consider the function 1/n → 1/pn which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parametrised family of Cramér type estimates on the gaps between successive primes. Here the parametrisation comes from the Hölder exponent. In particular, we show that Cramér’s conjecture is equivalent to the map 1/n → 1/pn being Lipschitz. On the other hand, we show that the inverse map 1/pn → 1/n is Hölder of all orders but not Lipschitz and this is independent of Cramér’s conjecture.

KW - Prime

KW - Prime gaps

KW - Cramér’s conjecture

KW - Hölder maps

KW - Lipschitz maps

U2 - 10.14321/realanalexch.46.2.0523

DO - 10.14321/realanalexch.46.2.0523

M3 - Article

SN - 0147-1937

VL - 46

SP - 523

EP - 532

JO - Real Analysis Exchange

JF - Real Analysis Exchange

IS - 2

ER -

On Hölder maps and prime gaps

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Keywords

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Projects

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this