Preprint Article Version 4 NOT YET PEER-REVIEWED

Farey Sequences and the Riemann Hypothesis

  1. Grayson College, P.O. Box 2171, Denison, TX 75021, USA
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 (doi: 10.20944/preprints201607.0003.v4). Cox, D. Farey Sequences and the Riemann Hypothesis. Preprints 2016, 2016070003 (doi: 10.20944/preprints201607.0003.v4).

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.

Subject Areas

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

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