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dc.contributor.authorBroughan, Kevin A.
dc.contributor.authorBarnett, A. Ross
dc.date.accessioned2009-01-16T03:04:31Z
dc.date.available2009-01-16T03:04:31Z
dc.date.issued2006
dc.identifier.citationBroughan, K. A. & Barnett, A. R.(2006). Linear law for the logarithms of the Riemann periods at simple critical zeta zeros. Mathematics of Computation. 75, 891-902.en
dc.identifier.urihttps://hdl.handle.net/10289/1795
dc.description.abstractEach simple zero 1/2 + iγn of the Riemann zeta function on the critical line with γn > 0 is a center for the flow s˙ = ξ(s) of the Riemann xi function with an associated period Tn. It is shown that, as γn →∞, log Tn ≥ π/4 γn + O(log γn). Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γn+1 − γn≥ γn-θ for some exponent θ > 0, we obtain the upper bound log Tn ≤ γn2 + θ Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log Tn = π/ 4 γn +O(log γn). Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.en
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherAmerican Mathematical Societyen
dc.relation.urihttp://www.ams.org/mcom/2006-75-254/S0025-5718-05-01803-X/home.htmlen
dc.rightsFirst published in Mathematics of Computation in volume 75, pages 891-902, published by the American Mathematical Society. ©2006 American Mathematical Society.en
dc.subjectmathematicsen
dc.titleLinear law for the logarithms of the Riemann periods at simple critical zeta zerosen
dc.typeJournal Articleen
dc.identifier.doi10.1090/S0025-5718-05-01803-X
dc.relation.isPartOfMathematics of Computationen_NZ
pubs.begin-page891en_NZ
pubs.elements-id31709
pubs.end-page902en_NZ
pubs.issue254en_NZ
pubs.volume75en_NZ
uow.identifier.article-no254en_NZ


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