Correlations among the Riemann zeros: Invariance, resurgence, prophecy and self-duality

Details Correlations among the Riemann zeros: Invariance, resurgence, prophecy and self-duality Correlations among the Riemann zeros: Invariance, resurgence, prophecy and self-duality

2004-05-25

arxiv.org/abs/nlin/0405058v2

Nonlinear Sciences

Chaotic Dynamics

16 pages, 10 figures

Scientific paper

We present a conjecture describing new long range correlations among the Riemann zeros leading to 3 principal features: (i) The spectral auto-correlation is invariant w.r.t. the averaging window. (ii) Resurgence occurs wherein the lowest zeros appear in all auto-correlations. (iii) Suitably defined correlations lead to predictions (prophecy) of new zeros. This conjecture is supported by analytical arguments and confirmed by numerical calculations using 10^{22} zeros computed by Odlyzko. The results lead to a self-duality of the Riemann spectrum similar to the quantum-classical duality observed in billiards.

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