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Article

Farey Sequences and the Riemann Hypothesis

This version is not peer-reviewed.

Submitted:

14 July 2016

Posted:

15 July 2016

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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 open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

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