Tuck's incompressibility function: statistics for zeta zeros and eigenvalues

Details Tuck's incompressibility function: statistics for zeta zeros and eigenvalues Tuck's incompressibility function: statistics for zeta zeros and eigenvalues

2008-07-22

arxiv.org/abs/0807.3474v1

Physics

Condensed Matter

Statistical Mechanics

40 Pages, 6 figures

Scientific paper

For any function that is real for real x, positivity of Tuck's function Q(x)=D'^2(x)/(D'^2(x)-D"(x) D(x)) is a condition for the absence of the complex zeros close to the real axis. Study of the probability distribution P(Q), for D(x) with N zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of Q are very rare for the Riemann zeros. P(Q) has singularities at Q=0, Q=1 and Q=N. The moments (averages of Q^m) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-N limit of P(Q) can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at Q=1, while the large-Q decay is determined by the pole closest to the origin. Determining the large-N limit of P(Q) for the GUE seems difficult.

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