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dc.contributor.authorSchleich, Wolfgang P.
dc.contributor.authorBezděková, Iva
dc.contributor.authorKim, Moochan B.
dc.contributor.authorAbbott, Paul C.
dc.contributor.authorMaier, Helmut
dc.contributor.authorMontgomery, Hugh L.
dc.contributor.authorNeuberger, John W.
dc.date.accessioned2018-06-11T11:35:17Z
dc.date.available2018-06-11T11:35:17Z
dc.date.issued2018
dc.identifier.citationPhysica Scripta. 2018, vol. 93, no. 6, art. no. 065201.cs
dc.identifier.issn0031-8949
dc.identifier.issn1402-4896
dc.identifier.urihttp://hdl.handle.net/10084/127295
dc.description.abstractWe prove the equivalence of three formulations of the Riemann hypothesis for functions f defined by the four assumptions: (a1) f satisfies the functional equation f (1 - s) = f (s) for the complex argument s = sigma + i tau, (a2) f is free of any pole, (a3) for large positive values of s the phase. of f increases in a monotonic way without a bound as tau increases, and (a4) the zeros of f as well as of the first derivative f ' of f are simple zeros. The three equivalent formulations are: (R1) All zeros of f are located on the critical line sigma = 1/2, (R2) All lines of constant phase theta of f corresponding to +/-pi, +/- 2 pi, +/- 3 pi, ... merge with the critical line, and (R3) All points where f' vanishes are located on the critical line, and the phases of f at two consecutive zeros of f' differ by pi. Our proof relies on the topology of the lines of constant phase of f dictated by complex analysis and the assumptions (a1)-(a4). Moreover, we show that (R2) implies (R1) even in the absence of (a4). In this case (a4) is a consequence of (R2).cs
dc.language.isoencs
dc.publisherIOP Publishingcs
dc.relation.ispartofseriesPhysica Scriptacs
dc.relation.urihttps://doi.org/10.1088/1402-4896/aabca9cs
dc.rights© 2018 IOP Publishing Ltdcs
dc.subjectRiemann hypothesiscs
dc.subjectlines of constant phasecs
dc.subjectcontinuous Newton methodcs
dc.titleEquivalent formulations of the Riemann hypothesis based on lines of constant phasecs
dc.typearticlecs
dc.identifier.doi10.1088/1402-4896/aabca9
dc.type.statusPeer-reviewedcs
dc.description.sourceWeb of Sciencecs
dc.description.volume93cs
dc.description.issue6cs
dc.description.firstpageart. no. 065201cs
dc.identifier.wos000433131400001


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