Linear law for the logarithms of the Riemann periods at simple critical zeta zeros

Broughan, Kevin A.; Barnett, A. Ross

Linear law for the logarithms of the Riemann periods at simple critical zeta zeros

dc.contributor.author	Broughan, Kevin A.
dc.contributor.author	Barnett, A. Ross
dc.date.accessioned	2009-01-16T03:04:31Z
dc.date.available	2009-01-16T03:04:31Z
dc.date.issued	2006
dc.description.abstract	Each 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.mimetype	application/pdf
dc.identifier.citation	Broughan, 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.doi	10.1090/S0025-5718-05-01803-X
dc.identifier.uri	https://hdl.handle.net/10289/1795
dc.language.iso	en
dc.publisher	American Mathematical Society	en
dc.relation.isPartOf	Mathematics of Computation	en_NZ
dc.relation.uri	http://www.ams.org/mcom/2006-75-254/S0025-5718-05-01803-X/home.html	en
dc.rights	First published in Mathematics of Computation in volume 75, pages 891-902, published by the American Mathematical Society. ©2006 American Mathematical Society.	en
dc.subject	mathematics	en
dc.title	Linear law for the logarithms of the Riemann periods at simple critical zeta zeros	en
dc.type	Journal Article	en
pubs.begin-page	891	en_NZ
pubs.elements-id	31709
pubs.end-page	902	en_NZ
pubs.issue	254	en_NZ
pubs.volume	75	en_NZ
uow.identifier.article-no	254	en_NZ