Newton flow of the Riemann zeta function: separatrices control the appearance of zeros

peer-reviewed
Erstveröffentlichung
2014-10-14Authors
Neuberger, J. W.
Feiler, Cornelia
Maier, Helmut
Schleich, Wolfgang P.
Wissenschaftlicher Artikel
Published in
New Journal of Physics ; 16 (2014), 10. - Art.-Nr. 103023. - eISSN 1367-2630
Link to original publication
https://dx.doi.org/10.1088/1367-2630/16/10/103023Faculties
Fakultät für Mathematik und WirtschaftswissenschaftenFakultät für Naturwissenschaften
Institutions
Institut für QuantenphysikInstitut für Zahlentheorie und Wahrscheinlichkeitstheorie
Document version
published version (publisher's PDF)Abstract
Abstract
A great many phenomena in physics can be traced back to the zeros of a function or a functional. Eigenvalue or variational problems prevalent in classical as well as quantum mechanics are examples illustrating this statement. Continuous descent methods taken with respect to the proper metric are efficient ways to attack such problems. In particular, the continuous Newton method brings out the lines of constant phase of a complex-valued function. Although the patterns created by the Newton flow are reminiscent of the field lines of electrostatics and magnetostatics they cannot be realized in this way since in general they are not curl-free. We apply the continuous Newton method to the Riemann zeta function and discuss the emerging patterns emphasizing especially the structuring of the non-trivial zeros by the separatrices. This approach might open a new road toward the Riemann hypothesis.
Subject headings
[GND]: Riemannsche Zetafunktion | Newton-Verfahren[LCSH]: Functions, Zeta | Newton-Raphson method
[Free subject headings]: Riemann zeta function | continuous Newton method | Newton flow
[DDC subject group]: DDC 530 / Physics
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Please use this identifier to cite or link to this item: http://dx.doi.org/10.18725/OPARU-46346
Neuberger, J. W. et al. (2022): Newton flow of the Riemann zeta function: separatrices control the appearance of zeros. Open Access Repositorium der Universität Ulm und Technischen Hochschule Ulm. http://dx.doi.org/10.18725/OPARU-46346
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