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On the exact location of the non-trivial zeros of Riemann’s zeta function

 

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Opened Access On the exact location of the non-trivial zeros of Riemann’s zeta function
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Author: Arias de Reyna Martínez, Juan
Van de Lune, Jan
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2014
Published in: Acta Arithmetica, 163 (3), 215-245.
Document type: Article
Abstract: In this paper we introduce the real valued real analytic function κ(t) implicitly defined by e 2πiκ(t) = −e −2iϑ(t) ζ 0 ( 1 2 − it) ζ 0( 1 2 + it) , (κ(0) = − 1 2 ). By studying the equation κ(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann’s ζ(s) and ζ 0 (s). Assuming the Riemann hypothesis and the simplicity of the zeros of ζ(s), it will follow that the ordinate of the zero 1/2 + iγn of ζ(s) will be the unique solution to the equation κ(t) = n.
Cite: Arias de Reyna Martínez, J. y Van de Lune, J. (2014). On the exact location of the non-trivial zeros of Riemann’s zeta function. Acta Arithmetica, 163 (3), 215-245.
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URI: http://hdl.handle.net/11441/43000

DOI: 10.4064/aa163-3-3

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