bibkey: burnol2003analyticestimate authors: Jean-François Burnol year: 2003 title: On an analytic estimate in the theory of the Riemann Zeta function and a Theorem of Baez-Duarte doi: null url: https://arxiv.org/abs/math/0202166v1 claim: Lemma 4.3 unconditionally identifies a damped Mobius fractional-part source on Re(s)>1; critical-line ratio control assumes RH, and the small-shift Hardy projection identity of Theorem 4.3 is RH-equivalent. strata_touched: [] license: citation-only triage: anchor

Damped Mellin sources and the separate Hardy condition

The inspected primary version is arXiv:math/0202166v1, a nine-page preprint dated 18 February 2002. Its arXiv metadata gives the journal reference Acta Cientifica Venezolana 54 (2003), 210–215, and no DOI. The journal text was not separately inspected. Statements below use the preprint’s numbering. The inspected statements are reused directly; no full proof audit or Lean certification is claimed.

Unconditional source identification

Section 4 uses $\{t\}=t-\lfloor t\rfloor$ and

$$f_{\epsilon }(u)=\sum _{n\ge 1}\mu (n)n^{-\epsilon }\left\{\frac{1}{nu}\right\},\epsilon >0.$$

Lemma 4.3, printed p.6, gives for $\Re s>1$

$${\int }_0^1{f}_{\epsilon }(u)u^{s-1}du=\frac{1}{\zeta (1+\epsilon )(s-1)}-\frac{\zeta (s)}{s\zeta (s+\epsilon )}.$$

This identity supplies a damped arithmetic source, a rational principal part and a one-sided Mellin support relation. It does not require RH. Theorem 4.1, printed pp.6–7, identifies the critical-line $L^2$ Mellin transform under the additional hypothesis $f_{\epsilon }\in L^2((0,\infty ),du)$; that hypothesis cannot be omitted. Theorem 4.2 says square integrability for a positive shift sequence tending to zero implies RH.

Conditions on critical-line transport

The paper’s uniform critical-line bound for $\zeta (s)/\zeta (s+A)$ explicitly assumes RH. Theorem 4.3, printed pp.7–8, states an RH equivalence for the left Hardy projection identity of

$$\frac{\zeta (s-\epsilon /2)}{\zeta (s+\epsilon /2)}\frac{1}{s-\epsilon /2},\Re s=1/2,$$

for small positive shifts; a sequence tending to zero suffices. Its projection is the explicitly stated rational term on $\Re z<1/2$. This is a substantive support condition, not a consequence of the boundary modulus. The source’s causality discussion explicitly distinguishes equal modulus from equal phase.

Actual derivative-source parameter map

In the FIB boundary calculation, §§404, 408, 411–413 use

$${\mathcal{G}}_{\epsilon }(z)=\frac{\mathcal{B}(z+\epsilon )}{\mathcal{B}(z)}\frac{-{\zeta }^{\prime }(z)}{z\zeta (z+\epsilon )}.$$

Set $R_{\epsilon }(z)=\zeta (z)/[z\zeta (z+\epsilon )]$. On a domain where these functions are defined, the direct algebraic map is

$${\mathcal{G}}_{\epsilon }(z)=-\frac{\mathcal{B}(z+\epsilon )}{\mathcal{B}(z)}\left({\partial }_z-{\partial }_{\epsilon }+\frac{1}{z}\right)R_{\epsilon }(z).$$

Burnol’s $s$ maps to $z$ and his shift is the same $\epsilon$. The displayed differential operator acts on $R_{\epsilon }$, not on the $\mathcal{B}$ multiplier. This parameter adaptation is repo-derived algebra, not a new derivative theorem or source estimate. Transporting the Mellin source through derivatives requires control of the introduced logarithmic weights; the displayed identity supplies no such uniform estimate.

The FIB rational correction $Q_{\epsilon }$ has a pole at $z=2$ even though its full weighted pairing is zero. A right-half-plane Hardy argument must separate that known rational component. For each fixed nontrivial zero $\rho$, all sufficiently small positive $\epsilon$ preserve the pole at $z=\rho -\epsilon$ with the original multiplicity; no common threshold over all zeros is asserted. Neither a null pairing nor a change of FIB coordinates proves the required holomorphy or the signed critical estimate.

The complete centered absolute norm and zero-damping boundary interface in §§411–413 are applications to the actual source with separately supplied domination. They are not statements printed in this paper. This note supplies the relevant existing Mellin/Hardy structure and its conditions; it supplies no unconditional proof of RH or full Robin.