Farey Sequences and the Riemann Hypothesis

Darrell Cox

doi:10.20944/preprints201607.0003.v2

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preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints201607.0003.v2

Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

Farey Sequences and the Riemann Hypothesis

Darrell Cox

Version 1 : Received: 4 July 2016 / Approved: 4 July 2016 / Online: 4 July 2016 (09:56:15 CEST)
Version 2 : Received: 6 July 2016 / Approved: 7 July 2016 / Online: 7 July 2016 (11:09:42 CEST)
Version 3 : Received: 14 July 2016 / Approved: 15 July 2016 / Online: 15 July 2016 (02:46:54 CEST)
Version 4 : Received: 17 July 2016 / Approved: 18 July 2016 / Online: 18 July 2016 (08:48:07 CEST)
Version 5 : Received: 23 July 2016 / Approved: 23 July 2016 / Online: 23 July 2016 (12:27:42 CEST)
Version 6 : Received: 9 August 2016 / Approved: 10 August 2016 / Online: 10 August 2016 (11:35:25 CEST)

How to cite: Cox, D. Farey Sequences and the Riemann Hypothesis. Preprints 2016, 2016070003. https://doi.org/10.20944/preprints201607.0003.v2 Cox, D. Farey Sequences and the Riemann Hypothesis. Preprints 2016, 2016070003. https://doi.org/10.20944/preprints201607.0003.v2

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MDPI and ACS Style

Cox, D. Farey Sequences and the Riemann Hypothesis. Preprints 2016, 2016070003. https://doi.org/10.20944/preprints201607.0003.v2

APA Style

Cox, D. (2016). Farey Sequences and the Riemann Hypothesis. Preprints. https://doi.org/10.20944/preprints201607.0003.v2

Chicago/Turabian Style

Cox, D. 2016 "Farey Sequences and the Riemann Hypothesis" Preprints. https://doi.org/10.20944/preprints201607.0003.v2

Abstract

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.

Keywords

Riemann hypothesis, Farey sequence, Gauss sum associated with a Dirichlet character

Subject

Computer Science and Mathematics, Algebra and Number Theory

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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