bibkey: coffey2006baezduarteextensions authors: “Mark W. Coffey” year: 2006 title: “On the coefficients of the Baez-Duarte criterion for the Riemann hypothesis and their extensions” doi: null url: “https://arxiv.org/abs/math-ph/0608050v2” claim: “Parametrized Baez-Duarte coefficients and their ordinary generating-function binomial transform are established prior tools; generalizing coefficient notation does not provide critical signed decay.” strata_touched: [] license: “Citation only; no source text is reproduced.” triage: anchor

Parametrized coefficients and binomial transforms

The retained primary version is arXiv:math-ph/0608050v2, updated 2006-09-07T19:52:20Z. Its 24-page PDF has SHA256 031aae16827824dbf864c790e82bda6a5b7dd3f27081c8234d7da932712c7b85. The versioned arXiv metadata supplies no DOI or journal reference. PDF page locators were checked against the source TeX; the PDF extraction reports missing font-encoding support, so mathematical symbols are read from the TeX source.

Equation (1) retains the original coefficients

$$c_k=\sum _{j=0}^k(-1{)}^j(\genfrac{}{}{0ex}{}{k}{j})\frac{1}{\zeta (2j+2)}.$$

Proposition 7, PDF page 17, equation (63), records the ordinary generating-function transformation for the parametrized reciprocal-Hurwitz coefficients $c_k(b,a)$, defined by equation (19). Its proof, equation (65), reorders their finite binomial sums into a generating series. This is an existing tool, not a new FIB result. The paper’s stated generating-function domain is retained as a source statement; this note does not independently certify analytic continuation throughout that domain.

Proposition 4, PDF pages 11–12, equations (41)–(43), treats a general Dirichlet series with a factor $(pj+p-1)$ in its coefficients and a factor $1/(s-1)$ in the expansion. Those factors distinguish that representation from the original reciprocal-zeta Newton coefficients. It cannot be applied by dropping either factor. The general analytic-function proposal immediately preceding Proposition 4 is explicitly labeled Conjecture 1.

For the project’s actual $e=\mu \ast \beta$, the coefficient at index $k$ is formed from the same signed kernel $n^{-2}(1-n^{-2}{)}^k$. The passage from $n=dm$ to a finite binomial average of the original $c_j$ is a source-scale application of the classical binomial theorem. A bound for that average must still be supplied with its index weights, and the complete $d$ tail must be retained. Neither this paper’s generalized notation nor its generating transform supplies unconditional $k^{-3/4+\epsilon }$ decay for the actual FIB coefficients.

The original RH criterion is reused from Báez-Duarte 2003. This source note makes no claim of a new criterion, exhaustive literature coverage, or originality of binomial transforms. It records primary-source scope rather than a Lean verification of the source paper.