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

TL;DR

This paper investigates Tuck's incompressibility function for zeros of the Riemann zeta function and GUE eigenvalues, revealing rare large values and differences in distribution compared to uncorrelated zeros, with implications for zero absence conditions.

Contribution

It introduces a detailed analysis of Tuck's function distribution for GUE eigenvalues and Riemann zeros, highlighting singularities and differences from Poisson distributions.

Findings

Large values of Q are rare for Riemann zeros.

P(Q) shows singularities at Q=0, 1, and N.

Moments of Q are smaller for GUE than Poisson zeros.

Abstract

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.