On the exact location of the non-trivial zeros
of Riemann's zeta function
Acta Arithmetica, Tome 163 (2014) no. 3, pp. 215-245
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
We introduce the real valued real analytic function $\kappa (t)$ implicitly defined by \[ e^{2\pi i \kappa (t)}=-e^{-2i\vartheta (t)}\frac {\zeta '(1/2-it)}{\zeta '(1/2+it)} \hskip 1em (\kappa (0)=-1/2).\] By studying the equation $\kappa (t) = n$ (without making any unproved hypotheses), we show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's $\zeta (s)$ and $\zeta '(s)$. Assuming the Riemann hypothesis and the simplicity of the zeros of $\zeta (s)$, it follows that the ordinate of the zero $1/2 + i \gamma _n$ of $\zeta (s)$ is the unique solution to the equation $\kappa (t) = n$.
Keywords:
introduce real valued real analytic function kappa implicitly defined kappa e vartheta frac zeta it zeta hskip kappa studying equation kappa without making unproved hypotheses function closely related exact position zeros riemanns zeta zeta assuming riemann hypothesis simplicity zeros zeta follows ordinate zero gamma zeta unique solution equation kappa
Affiliations des auteurs :
Juan Arias de Reyna 1 ; Jan van de Lune 2
@article{10_4064_aa163_3_3,
author = {Juan Arias de Reyna and Jan van de Lune},
title = {On the exact location of the non-trivial zeros
of {Riemann's} zeta function},
journal = {Acta Arithmetica},
pages = {215--245},
publisher = {mathdoc},
volume = {163},
number = {3},
year = {2014},
doi = {10.4064/aa163-3-3},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa163-3-3/}
}
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Juan Arias de Reyna; Jan van de Lune. On the exact location of the non-trivial zeros of Riemann's zeta function. Acta Arithmetica, Tome 163 (2014) no. 3, pp. 215-245. doi: 10.4064/aa163-3-3
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