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

##### Abstract

Each simple zero 1/2 + iÎ³n of the Riemann zeta function on the critical line with Î³n > 0 is a center for the flow sË™ = Î¾(s) of the Riemann xi function with an associated period Tn. It is shown that, as Î³n â†’âˆž, log Tn â‰¥ Ï€/4 Î³n + O(log Î³n).
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 Î³n+1 âˆ’ Î³nâ‰¥ Î³n-Î¸ for some exponent Î¸ > 0, we obtain the upper bound log Tn â‰¤ Î³n2 + Î¸ 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 = Ï€/ 4 Î³n +O(log Î³n). 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 Hilbertâ€“Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.

##### Type

Journal Article

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##### Series

##### 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.

##### Date

2006

##### Publisher

American Mathematical Society

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##### Rights

First published in Mathematics of Computation in volume 75, pages 891-902, published by the American Mathematical Society. Â©2006 American Mathematical Society.