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

2016-07-012016-07-012014Arias 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), 215-245.0065-10361730-6264http://hdl.handle.net/11441/43000In 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.application/pdfengAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Zeta functionNon-trivial zerosDistribution of zerosOn the exact location of the non-trivial zeros of Riemann’s zeta functioninfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccess10.4064/aa163-3-3https://idus.us.es/xmlui/handle/11441/43000