bibkey: cislo2008riesz authors: Jerzy Cislo and Marek Wolf year: 2008 title: On the Riesz and Baez-Duarte criteria for the Riemann Hypothesis doi: null url: https://arxiv.org/abs/0807.2971v1 claim: The Riesz function and the finite Baez-Duarte sequence have an exponential generating identity and signed Moebius series representations. strata_touched:
- D5/S3/Weil/ZetaBridge/RieszBaezDuarte license: citation-only triage: anchor
Riesz and Baez-Duarte Transforms
Equations (2) and (4) define the Riesz function and the discrete coefficients:
[ R(x)=x\sum_{j=0}^{\infty}\frac{(-1)^j x^j}{j!,\zeta(2j+2)}, \qquad c_k=\sum_{j=0}^{k}\frac{(-1)^j\binom{k}{j}}{\zeta(2j+2)}. ]
The paper’s equation (10) is the exponential generating identity:
[ \sum_{k=0}^{\infty}\frac{x^k}{k!}c_k=\frac{e^x}{x}R(x). \tag{10} ]
Equations (13) and (14) give the signed Moebius series:
[ R(x)=x\sum_{j=0}^{\infty}\frac{(-x)^j}{j!,\zeta(2j+2)} =x\sum_{n=1}^{\infty}\frac{\mu(n)}{n^2}e^{-x/n^2}, \tag{13} ]
[ c_k=\sum_{j=0}^{k}\frac{\binom{k}{j}(-1)^j}{\zeta(2j+2)} =\sum_{n=1}^{\infty}\frac{\mu(n)}{n^2} \left(1-\frac{1}{n^2}\right)^k. \tag{14} ]
Here the dummy index in (2) has been renamed to j. The paper’s c_k is the repository theory’s b_k and the formalization’s baezDuarte k, with the same finite alternating binomial definition. The formal Riesz function has domain Real; its two HasSum identities are stated for real x > 0. These are the repository’s positive-real scope, not extra printed quantifiers attributed to equations (10) and (13). The discrete identity includes every natural k, including zero. Reindexing the positive arithmetic index as n+1 retains the entire signed Moebius series, including mu(1)=1. At k=0 the formalization uses the natural-power convention 0^0=1; the paper prints no separate convention for that endpoint.
At the positive even zeta arguments the actual complex zeta value is real and nonzero, so its reciprocal equals the real reciprocal used in the Lean definitions after the real-to-complex identification. The HasSum values are respectively exp(x)*(riesz x/x), riesz x/x, and baezDuarte k. Multiplying equation (10) by exp(-x) gives the theory’s R(x)/x presentation; multiplying the arithmetic Riesz HasSum equality by nonzero x recovers equation (13). HasSum certifies convergence as well as the value. The source calls its intermediate operator equation (8) formal and does not print a separate absolute-convergence calculation for these interchanges; the formalization supplies those arguments.
This note records literature and its notation correspondence. It adds no theorem, coverage, or priority claim. The identities alone do not establish the Lemma 3 error O(k^(-3/2)), Lemma 4’s uniform variation on all 0 < x < y, Theorem 1’s growth equivalence for delta > -3/2, or either original RH criterion with its quantifier over every epsilon > 0.
Source Validation
The authorized primary-source intake crosschecked PDF pages 2-3 against CW_xxx.tex in the pinned v1 source archive: definition (2), lines 84-88; definition (4), lines 99-104; equation (10), lines 154-158; equations (13)-(14), lines 178-190. This is inherited source-transcription evidence, not an independent proof review. The PDF and TeX name the first author Jerzy Cislo. The repository theory bibliography’s first name Jan is a bibliographic discrepancy.
The intake records arXiv v1 submission at 2008-07-18T13:23:33Z and observes
only v1 in the arXiv submission history. The PDF title page prints October
28, 2018; its creation and modification metadata read
D:20181030111744-04'00'. Those printed and PDF-generation dates do not
establish a later arXiv version. The citation year follows the 2008
submission. No DOI was verified, so the citation uses doi: null and the
exact stable HTTPS v1 URL.
Verified locator
- https://arxiv.org/abs/0807.2971v1
- https://arxiv.org/pdf/0807.2971v1
- https://arxiv.org/src/0807.2971v1
- Repository source, F09 and E14/E15: https://github.com/the-omega-institute/trureturing/blob/915a86bf19ec91fdbd690a70e75c84014d237b7d/docs/develop/theory/RH_RESEARCH_LANE_THEORY.md