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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Linear law for the logarithms of the Riemann periods at simple critical zeta zeros
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by Kevin A. Broughan and A. Ross Barnett PDF
Math. Comp. 75 (2006), 891-902 Request permission

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 $\gamma _n \rightarrow \infty$, \[ \log T_n\ge \frac {\pi }{4}\gamma _n+O(\log \gamma _n).\] 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
  • Received by editor(s): December 13, 2004
  • Received by editor(s) in revised form: March 17, 2005
  • Published electronically: November 30, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Math. Comp. 75 (2006), 891-902
  • MSC (2000): Primary 11M06, 11M26, 11S40
  • DOI: https://doi.org/10.1090/S0025-5718-05-01803-X
  • MathSciNet review: 2196998