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The holomorphic flow of the Riemann zeta function


Authors: Kevin A. Broughan and A. Ross Barnett
Journal: Math. Comp. 73 (2004), 987-1004
MSC (2000): Primary 30A99, 30C10, 30C15, 30D30, 32M25, 37F10, 37F75
DOI: https://doi.org/10.1090/S0025-5718-03-01529-1
Published electronically: November 26, 2003
Corrigendum: Math. Comp. 76 (2007), 2249-2250
MathSciNet review: 2031420
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Abstract | References | Similar Articles | Additional Information

Abstract: The flow of the Riemann zeta function, $\dot{s}=\zeta(s)$, is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica.

The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.


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Additional Information

Kevin A. Broughan
Affiliation: University of Waikato, Hamilton, New Zealand
Email: kab@waikato.ac.nz

A. Ross Barnett
Affiliation: University of Waikato, Hamilton, New Zealand
Email: arbus@waikato.ac.nz

DOI: https://doi.org/10.1090/S0025-5718-03-01529-1
Keywords: Dynamical system, phase portrait, critical point, orbit, separatrix, Riemann zeta function
Received by editor(s): April 7, 2002
Received by editor(s) in revised form: May 30, 2002
Published electronically: November 26, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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