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A boundary limit for Suzuki's arithmetic canonical system

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

For the shifted Riemann xi-function, we prove the large-radius boundary limit of Suzuki's arithmetic canonical system. The proof applies to every positive shift for which the reflected quotient is pole-free in the upper half-plane and the two entire functions have no common zero. It therefore applies unconditionally to shifts at least one half, including Suzuki's original range greater than one. We give the weak-kernel construction needed at the smaller shifts, two integral identities for its positive determinant ratio, and decay estimates faster than every fixed exponential in logarithmic radius. The hypotheses on the shift are retained throughout.

Provenance statement

The mathematical draft and exposition were produced primarily with OpenAI Codex under the direction of the listed contributors. Lean was used to check supporting entire-function, Fourier-operator, and analytic-estimate lemmas.

Tools used

OpenAI
Codex
Lean
Lean

References

  1. M. Suzuki, A canonical system of differential equations arising from the Riemann zeta-function , RIMS Kôkyûroku Bessatsu B34 (2012), 397--435; revised preprint, arXiv:1204.1827v2 (https://arxiv.org/abs/1204.1827v2) (2016).arXiv
  2. M. Suzuki, Chains of reproducing kernel Hilbert spaces generated by unimodular functions , Ann. Inst. Fourier (Grenoble) 75 (2025), no. 4, 1463--1508, doi:10.5802/aif.3705 (https://doi.org/10.5802/aif.3705); arXiv:2012.11121v2 (https://arxiv.org/abs/2012.11121v2).arXivDOI

Version history

  1. v1 · 9 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 11, 2026