We introduce the real valued real analytic function κ(t) implicitly defined by (κ(0) = -1/2). By studying the equation κ(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 ζ(s) and ζ’(s). Assuming the Riemann hypothesis and the simplicity of the zeros of ζ(s), it follows that the ordinate of the zero 1/2 + iγₙ of ζ(s) is the unique solution to the equation κ(t) = n.
@article{bwmeta1.element.bwnjournal-article-doi-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},
volume = {166},
year = {2014},
pages = {215-245},
zbl = {1301.11062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-3-3}
}
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 166 (2014) pp. 215-245. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-3-3/