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), 891902
MSC (2000):
Primary 11M06, 11M26, 11S40
Published electronically:
November 30, 2005
MathSciNet review:
2196998
Fulltext PDF Free Access
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Abstract: Each simple zero of the Riemann zeta function on the critical line with is a center for the flow of the Riemann xi function with an associated period . It is shown that, as , 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 for some exponent , we obtain the upper bound . 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, . 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 HilbertPolya conjecture. They may provide a goal for those seeking a selfadjoint 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:
http://dx.doi.org/10.1090/S002557180501803X
PII:
S 00255718(05)01803X
Keywords:
Riemann zeta function,
xi function,
zeta zeros,
periods,
critical line,
HilbertPolya 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
