## The holomorphic flow of the Riemann zeta function

##### Citation

Broughan, K.A. & Barnett, A.R.(2003). The holomorphic flow of the Riemann zeta function . Mathematics of Computation, 73, 987-1004.

Permanent Research Commons link: http://hdl.handle.net/10289/1263

##### Abstract

The flow of the Riemann zeta function, ś = ς(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.

##### Date

2003-11##### Type

##### Publisher

American Mathematical Society

##### Rights

First published in Mathematics of Computation in volume 73, pages 987-1004, published by the American Mathematical Society. Copyright 2003, American Mathematical Society.