Phase of the Riemann ζ function and the inverted harmonic oscillator

R. K. Bhaduri, Avinash Khare, and J. Law
Phys. Rev. E 52, 486 – Published 1 July 1995
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Abstract

The Argand diagram is used to display some characteristics of the Riemann ζ function. The zeros of the ζ function on the complex plane give rise to an infinite sequence of closed loops, all passing through the origin of the diagram. The behavior of the phase of the ζ function on and off the line of zeros is studied. Up to some distance from the line of the complex zeros, the phase angle is shown to still retain their memory. The Argand plots also lead to an analogy with the scattering amplitude and an approximate rule for the location of the zeros. The smooth phase of the ζ function along the line of the zeros is related to the quantum density of states of an inverted oscillator.

  • Received 2 December 1994

DOI:https://doi.org/10.1103/PhysRevE.52.486

©1995 American Physical Society

Authors & Affiliations

R. K. Bhaduri, Avinash Khare, and J. Law

  • Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1

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Vol. 52, Iss. 1 — July 1995

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