bibkey: pyvovarov2026baezduarteremarks authors: “Alexandre Pyvovarov” year: 2026 title: “A few remarks on the Baez-Duarte Criterion” doi: null url: “https://arxiv.org/abs/2607.12084v3” claim: “The preprint’s final section explicitly retains a global bilinear remainder estimate as unresolved; its truncated norm formulas do not supply an unconditional RH proof.” strata_touched: [] license: “Citation only; no source text is reproduced.” triage: anchor
Exponential approximants and the retained remainder
The retained primary version is arXiv:2607.12084v3, updated
2026-07-21T15:54:34Z. Its 66-page PDF has SHA256
c85660d6c2bcbca5458e82cb048afd8c62311c216fd1fe88cbea8d6ffb4c48c7.
The versioned arXiv metadata supplies no DOI or journal reference.
The source TeX and PDF page locators were inspected for the concluding
remainder statement. Mathematical symbols use the TeX source where PDF
extraction reports missing font-encoding support.
The preprint studies exponentially damped Möbius combinations of balanced floor-function probes. These are Hilbert approximation vectors, distinct from the original Newton coefficients even though both involve Báez-Duarte criteria. Its notation denotes a deleted analytic remainder, not the project’s atomic Fibonacci substitution .
Corollary 9.38, PDF page 63, lists four sufficient estimates in (9.39) and a single combined boundedness condition in (9.40). Section 9.6, PDF pages 63–64, explicitly states that (9.40) remains unresolved and that boundedness cannot be transported from the truncated quantity to the complete quantity before the total remainder is controlled. The source also distinguishes existing Möbius–cotangent power-saving estimates with other summation regions and weights from this required combined estimate.
This note records that explicit limitation as prior-art evidence. It does not certify every theorem or intermediate algebraic formula in the preprint, and none is imported as a Lean-verified result. A growing upper bound for a truncated norm is not the boundedness estimate required for the full approximation. Separately estimating the four components is sufficient but can be stronger than the combined condition, because cancellation can occur between them.
For the actual FIB transport, an available absolute bound can pay a source tail only at the scale it explicitly reaches. It cannot be used to declare a remaining signed principal sum bounded. The distinct natural-prefix obstruction and its subsequence boundary remain those of Báez-Duarte 2000. Neither this preprint nor the source-tail interface establishes full Robin, actual critical FIB growth, or RH. This versioned check is a bounded literature comparison, not an exhaustive account of all current research.