Prime-shift operators on Björner's complex of squarefree integers: an exact decomposition of the Mertens invariant, and a corrected analysis of the 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
2026-09-18
DOI
https://doi.org/10.5281/zenodo.22823015
Primary Topic
Advanced Algebra and Geometry
Type
preprint
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preprint

Prime-shift operators on Björner's complex of squarefree integers: an exact decomposition of the Mertens invariant, and a corrected analysis of the Toeplitz--Hankel pipeline for the Guinand--Weil explicit formula

Luca Eliseo Pavesi
Zenodo (CERN European Organization for Nuclear Research)
Advanced Algebra and Geometry
preprint

Prime-shift operators on Björner's complex of squarefree integers: an exact decomposition of the Mertens invariant, and a corrected analysis of the 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 $c_{p}$. This paper is the fourth revision of \cite{pavesi2026v1}, superseding the first three. It contains both corrections of errors and new exact results. Corrections. (i) The bipartite block $A: C_{\mathrm{even}} \to C_{\mathrm{odd}}$ of $H_{n}$ is not $\sum_{p\nmid m}c_{p}e_{pm}$ but the full sum$Ae_{m} = \sum_{p\nmid m,pm\le n}c_{p}e_{pm} + \sum_{p\mid m}c_{p}e_{m/p}$,which we write as $A = A_{\uparrow} + A_{\downarrow}$. (ii) The maximal elements span $\ker A_{\uparrow}$, not $\ker A$; in fact $\ker A = \{0\}$ for $n\ge 2$. (iii) The values $c_{*}$ reported in the earlier revisions correspond to $A_{\uparrow}$, not to $A$; at $n=10^{4}$, $\operatorname{rank}(A_{\uparrow})=1125$, $\operatorname{rank}(A)=2141$, so $c_{*}=0.185$ while the full density is $c=0.352$. (iv) The kernel of $H_{n}$ therefore has density $1-2c\approx 0.296$, not $1-2c_{*}\approx 0.630$. New exact results. (v) We prove that the maximal elements of $[1,n]$ are exactly $m = P_{k}m'$ with $P_{k}=p_{1}\cdots p_{k}$ the $k$-th primorial, $m'$ squarefree coprime to $P_{k}$, and $m' \in (n/(p_{k+1}P_{k}),\, n/(p_{k}P_{k})]$. This yields the exact formula\[ M_{\mathcal{M}}(n) \;=\; \sum_{k\ge 0}(-1)^{k}\Bigl[T_{k}\!\Big(\Big\lfloor\tfrac{n}{p_{k}P_{k}}\Big\rfloor\Big) - T_{k}\!\Big(\Big\lfloor\tfrac{n}{p_{k+1}P_{k}}\Big\rfloor\Big)\Bigr], \qquad T_{k}(X) = \sum_{\substack{d\le X\\ d \text{ } P_{k}\text{-smooth}}}M(\lfloor X/d\rfloor).\](vi) We prove the identity $M(n) = M_{\mathcal{M}}(n) + M_{\mathcal{M}}^{T}(n)$ is not the correct decomposition; instead$M(n) = M_{\mathcal{M}}(n) + \dim R_{n}^{\uparrow} - \dim R_{n}^{\uparrow,T}$,where $R_{n}^{\uparrow} = \ker A_{\uparrow}|_{C_{\mathrm{even}}} \ominus \mathcal{M}$ is the residual of relations. (vii) We observe numerically that$|M_{\mathcal{M}}(n)|/\sqrt{n}$ oscillates in the range $[0.05,0.40]$ for $n\le 2\times 10^{5}$, with typical value near $0.29\approx 1-1/\sqrt{2}$, consistent with $M_{\mathcal{M}}(n) \sim (1-1/\sqrt{2})\sqrt{n}$ in RMS under RH. \textbf{What this paper does \emph{not} do.} We do not obtain a new characterisation of RH. The decomposition $M(n) = M_{\mathcal{M}}(n) + (\dim R_{n}^{\uparrow} - \dim R_{n}^{\uparrow,T})$ is algebraic: every term is $\Theta(\sqrt{n})$ in RMS under RH, and any unconditional bound of the form $M_{\mathcal{M}}(n) = O(\sqrt{n})$ would already imply RH by telescoping over dyadic intervals. We explain in Sref{sec:no-new-char} why this is unavoidable, and we formulate the spectral modification $\tilde H_{n}$ of $H_{n}$ as an open conjecture. No proof of the Riemann Hypothesis is claimed.

Zenodo (CERN European Organization for Nuclear Research)
Advanced Algebra and Geometry
DOIOpen Access PDFOpenAlex
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