Pair and triple correlations of Riemann zeros at low heights: limits of effective-size corrections

Abstract

To leading order, the Riemann zeros at height T deviate from the random-matrix limit in the same way as the eigenvalues of a random unitary matrix of finite size N_e ∝ log T. For the zeros studied here, this effective matrix is tiny: N_e is about 1 to 5. We ask whether so small a matrix still describes the corrections, and whether one N_e serves both pairs and triples of zeros. Using about 1.5×10⁷ zeros from Odlyzko's and Platt's tables, we compare three effective-size approximations with the full arithmetic formulas of Bogomolny and Keating (pairs) and Conrey and Snaith (triples). The plain correction fits the data worse than no correction at all. With the internal rescaling of Bornemann, Forrester and Mays, it improves the pair correlation at the two upper heights but worsens the triple correlation over a comparable range of separations. Two exploratory checks examine this discrepancy. Weighting the pair separations like the triple spans removes most of the pair improvement but does not reproduce the deterioration for triples. Splitting off the connected, genuinely three-point part of the triple correlation shows that this part is fitted better than by the sine kernel: the extra error arises outside it, although rules fixed in advance do not single out its source. The full formulas remain closer to the data than every approximation tested. This record contains the note (PDF) and an archive with the analysis code, the frozen protocol (in Polish) with its reporting annex and a dated English chronology, the six input archives, the stage records, the SHA-256 manifests of inputs and results, the scripts that print every table and draw the figures, the two exploratory analyses with an English description of their rules and their logs, and a script that checks every file against the manifests. The large results archives are regenerated by the scripts (see README.md in the archive). Version 1.1: the note is rewritten under a new title; the count of analysed zeros is corrected to about 1.48×10⁷; two exploratory analyses added after the main results (a comparison at matched total span and a split of the triple correlation into its connected and pair-inherited parts) are included with their code and logs. The archive's LICENSE now states the licences of this deposit (MIT for code, CC BY 4.0 for everything else); apart from README.md, SHA256SUMS.txt and LICENSE, the files of the version 1.0 archive are unchanged.