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


Authors: Kevin A. Broughan and A. Ross Barnett
Journal: Math. Comp. 75 (2006), 891-902
MSC (2000): Primary 11M06, 11M26, 11S40
DOI: https://doi.org/10.1090/S0025-5718-05-01803-X
Published electronically: November 30, 2005
MathSciNet review: 2196998
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Abstract: Each simple zero $ \frac{1}{2}+i\gamma_n$ of the Riemann zeta function on the critical line with $ \gamma_n > 0$ is a center for the flow $ \dot{s}=\xi(s)$ of the Riemann xi function with an associated period $ T_n$. It is shown that, as $ \gamma_n \rightarrow\infty$,

$\displaystyle \log T_n\ge \frac{\pi}{4}\gamma_n+O(\log \gamma_n).$

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 $ \gamma_{n+1}-\gamma_n \gg \gamma_n^{-\theta}$ for some exponent $ \theta>0$, we obtain the upper bound $ \log T_n \ll \gamma^{2+\theta}_n$. 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 T_n = \frac{\pi}{4}\gamma_n+O(\log \gamma_n)$. 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.


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Additional Information

Kevin A. Broughan
Affiliation: Department of Mathematics, University of Waikato, Hamilton, New Zealand
Email: kab@waikato.ac.nz

A. Ross Barnett
Affiliation: Department of Mathematics, University of Waikato, Hamilton, New Zealand
Email: arbus@math.waikato.ac.nz

DOI: https://doi.org/10.1090/S0025-5718-05-01803-X
Keywords: Riemann zeta function, xi function, zeta zeros, periods, critical line, Hilbert--Polya conjecture
Received by editor(s): December 13, 2004
Received by editor(s) in revised form: March 17, 2005
Published electronically: November 30, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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