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**1 - 1**of**1**### LINEAR LAW FOR THE LOGARITHMS OF THE RIEMANN PERIODS AT SIMPLE CRITICAL ZETA ZEROS

"... 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) oftheRiemannxi function with an associated period Tn. Itisshownthat,asγn→∞, log Tn ≥ π 4 γn + O(log γn). Numerical evaluation leads to the conjecture that this inequal ..."

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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) oftheRiemannxi function with an associated period Tn. Itisshownthat,asγn→∞, 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 ≫ γ−θ n for some exponent θ>0, we obtain. 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, the upper bound log Tn ≪ γ 2+θ n 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. 1.