dc.contributor.author Broughan, Kevin A. dc.date.accessioned 2009-02-18T01:06:53Z dc.date.available 2009-02-18T01:06:53Z dc.date.issued 2005 dc.identifier.citation Broughan, K.A. (2005). The holomorphic flow of Riemann’s function ξ(z). Nonlinearity, 18(3), 1269-1294. en dc.identifier.uri https://hdl.handle.net/10289/2027 dc.description.abstract The holomorphic flow z = ξ(z) of Riemann’s xi function is considered. Phase portraits are plotted and the following results, suggested by the portraits, proved: all separatrices tend to the positive and/or negative real axes. These are an infinite number of crossing separatrices. In the region between each pair of crossing separatrices- a band- there is at most one zero on the critical line. All zeros on the critical line are centres or have all elliptic sectors. The flows for ξ(z) and cosh (z) are linked with a differential equation. Simple zeros on the critical line and Gram points never coincide. The Riemann hypothesis is equivalent to all zeros being centres or multiple together with the non-existence of separatices which enter and leave a band in the same half plane. en dc.format.mimetype application/pdf dc.language.iso en dc.publisher Institute of Physics Publishing en_NZ dc.rights This article has been published in the journal: Nonlinearity. ©2005 IOP Publishing Ltd and London Mathematical Society. en dc.subject Riemann’s function Xi(s) en dc.title The holomorphic flow of Riemann’s function ξ(z) en dc.type Journal Article en dc.identifier.doi 10.1088/0951-7715/18/3/017 en dc.relation.isPartOf Nonlinearity en_NZ pubs.begin-page 1269 en_NZ pubs.elements-id 30732 pubs.end-page 1294 en_NZ pubs.issue 3 en_NZ pubs.volume 18 en_NZ
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