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

Juan Arias de Reyna, and Jan van de Lune. "On the exact location of the non-trivial zeros of Riemann's zeta function." Acta Arithmetica 163.3 (2014): 215-245. <http://eudml.org/doc/279194>.

@article{JuanAriasdeReyna2014,
abstract = {We introduce the real valued real analytic function κ(t) implicitly defined by $e^\{2πiκ(t)\} = -e^\{-2iϑ(t)\} (ζ^\{\prime \}(1/2-it))/(ζ^\{\prime \}(1/2+it))$ (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.},
author = {Juan Arias de Reyna, Jan van de Lune},
journal = {Acta Arithmetica},
keywords = {zeta function; non-trivial zeros; distribution of zeros},
language = {eng},
number = {3},
pages = {215-245},
title = {On the exact location of the non-trivial zeros of Riemann's zeta function},
url = {http://eudml.org/doc/279194},
volume = {163},
year = {2014},
}

TY - JOUR
AU - Juan Arias de Reyna
AU - Jan van de Lune
TI - On the exact location of the non-trivial zeros of Riemann's zeta function
JO - Acta Arithmetica
PY - 2014
VL - 163
IS - 3
SP - 215
EP - 245
AB - We introduce the real valued real analytic function κ(t) implicitly defined by $e^{2πiκ(t)} = -e^{-2iϑ(t)} (ζ^{\prime }(1/2-it))/(ζ^{\prime }(1/2+it))$ (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.
LA - eng
KW - zeta function; non-trivial zeros; distribution of zeros
UR - http://eudml.org/doc/279194
ER -