Extreme Spacings of Zeros: A Disanalogy Between Gaussian Analytic Functions and the Riemann Zeta Function

Abstract

This essay examines whether the extreme value statistics of zeros of Gaussian analytic functions (GAF), which converge to a Poisson point process with a universal $x^{3}e^{-x^{4}}$ density, could model the smallest gaps between critical zeros of the Riemann zeta function.

The Source Result

Feng and Yao study the minimal distances between zeros of Gaussian analytic functions on compact Riemann surfaces. For a random holomorphic section of a line bundle of degree $n$ (yielding $n$ zeros), they consider the $\binom{n}{2}$ pairwise geodesic distances $\sigma_{1} < \sigma_{2} < \dots$. Their main theorem establishes that after rescaling by $n^{3/4}$, the point process of these smallest distances converges to a Poisson point process on the line with intensity proportional to $x^{3}$. Consequently, the $k$-th smallest distance has a limiting density proportional to $x^{4k-1}e^{-x^{4}}$.

The Tempting Analogy

The Riemann Hypothesis is deeply connected to the statistics of zero spacings. Montgomery's pair correlation conjecture and Odlyzko's numerical investigations suggest that the local correlations of zeta zeros match those of eigenvalues of random Hermitian matrices (GUE). One might speculate that extreme value theory—studying the smallest gaps—could reveal a universal Poisson limit, suggesting that "exceptionally close" zeros are rare and essentially independent events, much like the GAF result.

Why the Bridge Collapses

This essay argues that the analogy is not viable. The source paper concerns a two-dimensional repulsive point process where the smallest distance scales as $n^{-3/4}$. In contrast, the critical zeros of $\zeta(s)$ form a one-dimensional process. For one-dimensional point processes with GUE statistics (the conjectured model for zeta), the smallest gap is expected to scale like $N^{-4/3}$ (where $N$ is the number of zeros), and its limiting distribution is not Poissonian but follows the GUE extreme-gap law. The exponents differ ($-3/4$ vs. $-4/3$), and the limiting laws are incompatible (Poisson vs. determinantal). Thus, the GAF model belongs to a different universality class than the zeta zeros.

Proposed Discriminating Test

We propose a statistical test using high-precision tables of zeta zeros. By computing the empirical distribution of the minimal gaps (rescaled by $(T\log T)^{3/4}$) and comparing it to the Feng-Yao density $x^{3}e^{-x^{4}}$, one would find a clear mismatch, confirming the disanalogy. The essay concludes by noting that a useful bridge would require a source paper on extreme spacings for the Sine process or other determinantal kernels conjecturally linked to zeta.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.