Prime-shift operators on Björner's complex of squarefree integers: spectral obstruction, kernel structure, and a Toeplitz--Hankel pipeline for the Guinand--Weil explicit formula

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-16
DOI
https://doi.org/10.5281/zenodo.22802275
Primary Topic
Algebraic structures and combinatorial models
Type
preprint
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preprint

Prime-shift operators on Björner's complex of squarefree integers: spectral obstruction, kernel structure, and a Toeplitz--Hankel pipeline for the Guinand--Weil explicit formula

Luca Eliseo Pavesi
Zenodo (CERN European Organization for Nuclear Research)
Algebraic structures and combinatorial models
preprint

Prime-shift operators on Björner's complex of squarefree integers: spectral obstruction, kernel structure, and a Toeplitz--Hankel pipeline for the Guinand--Weil explicit formula

Luca Eliseo Pavesi
preprint en

Abstract

Let $\Delta_{n}$ denote Bj\"orner's simplicial complex on the squarefree integers $m\leq n$ and let $H_{n} = \sum_{p\leq n}c_{p}(T_{p} + T_{p}^{*})$ be the self-adjoint operator obtained from the prime-shift operators $T_{p}$ with bounded weights $c_{p}$. We present a corrected analysis of $H_{n}$ and of the Toeplitz pencil constructed from the Guinand--Weil explicit formula. Four groups of results. First, we confirm the structural properties of $H_{n}$: self-adjointness, $\mathbb{Z}/2$-symmetric spectrum, the trace formula $\operatorname{Tr}(H_{n}^{2}) = 2\sum_{p\leq n}c_{p}^{2}\sigma_{p}(n/p)$, and a rigorous Carleman bound for the limiting measure when $c_{p}=p^{-s}$ with $s>1$. Second, we correct several statements of the earlier versions: the mixed products $T_{p}T_{q}^{*}$ and $T_{q}^{*}T_{p}$ do \emph{not} commute once the complex is truncated at $n$; the Mertens invariant is the \emph{index} of the bipartite block $A$, not the signature of $H_{n}$ (which vanishes by symmetry); and the kernel of the up-shift part $A_{\uparrow}$ restricted to $C_{1}$ has dimension $\pi(n)-\pi(n/2)$, whereas $\dim(\ker H_{n}\cap C_{1}) = \pi(n)-\pi(n/2)-1$ for $n\ge 15$. Third, we establish a structural decomposition of the kernel of $A$ into \emph{maximal elements} and a residual subspace of \emph{relations}, and we compute the density of the maximal part in closed form: it equals $c_{1}=\sum_{k\ge 0}(p_{k}^{-1}-p_{k+1}^{-1})\prod_{j\le k}(p_{j}+1)^{-1}=0.5675527\ldots$, in agreement with the numerics. The residual grows linearly in $n$ at a rate of approximately $0.02\,n$. Fourth, we revisit the Toeplitz pencil for the explicit formula. The prefactor of the prime-power term for the Gaussian test function is $1/(4\sigma\sqrt{\pi})$; the value $1/(8\pi\sigma\sqrt{\pi})$ proposed in the first revision is erroneous, and arose from a spurious factor $1/(2\pi)$ attached to the prime-power sum in the statement of the Guinand--Weil formula. We verify the correct identity numerically against the direct sum over the low-lying nontrivial zeros, with relative error below $10^{-13}$, while the competing prefactor fails by factors of order unity or larger already at $\Delta=\log 2$. We prove an exact-recovery theorem for the pencil and verify it numerically on synthetic data. We then discuss a Hankel--ESPRIT pipeline built from the corrected formula and identify the structural bottlenecks (effective rank and truncation of the prime-power sum) that govern how many zeros can be recovered. No proof of the Riemann Hypothesis is claimed.

Zenodo (CERN European Organization for Nuclear Research)
Algebraic structures and combinatorial models
DOIOpen Access PDFOpenAlex
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