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

• Autorzy: Juan Arias de Reyna , Jan van de Lune
• Języki publikacji: EN

Abstrakty

EN

We introduce the real valued real analytic function κ(t) implicitly defined by
$e^{2πiκ(t)} = -e^{-2iϑ(t)} (ζ'(1/2-it))/(ζ'(1/2+it))$ (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.

Kategorie tematyczne

• 26E05: Real-analytic functions
• 11M41: Other Dirichlet series and zeta functions
• 11M26: Nonreal zeros of ζ ( s ) and L ( s , χ ) ; Riemann and other hypotheses

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 3
• Strony: 215-245

Daty

wydano

2014

Twórcy

autor

Juan Arias de Reyna

• Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain

autor

Jan van de Lune

• Langebuorren 49, 9074 CH Hallum, The Netherlands, (formerly at the CWI, Amsterdam)

Identyfikatory

DOI

10.4064/aa163-3-3