bibkey: baezduarte2000natural authors: Luis Báez-Duarte year: 2000 title: Arithmetical Aspects of Beurling’s Real Variable Reformulation of the Riemann Hypothesis doi: null url: https://arxiv.org/abs/math/0011254v1 claim: “Proposition 4.4 excludes full-sequence L2 convergence of the unbalanced natural Möbius prefixes on the half-line. It does not exclude every subsequence or freely chosen coefficients in the Nyman–Beurling criterion.” strata_touched: [] license: citation-only triage: anchor

The natural-prefix obstruction and its exact scope

The pinned source is arXiv:math/0011254v1, submitted 2000-11-29. The 21-page PDF has SHA256 951ecc56eaae5b80eb8128ab4f98b5184d06ed97da608a130d65940daea8e94c. The inspected author TeX has SHA256 32453b451c900c7c557fc7da5eb8b93987d9d2addca4d31d293b9f844e84801d. The selected definitions, Proposition 4.4 and its proof, and the subsequent scope distinctions were read. This is source reuse, not a new proof or a Lean verification of the paper.

Space, coefficients and cutoff

Section 1 works in $L^p((0,\infty ),dx)$, defines $\chi =1_{(0,1]}$, and uses the fractional part $\rho (t)=t-\lfloor t\rfloor$. Its dilation is $K_{a}f(x)=f(ax)$. Equations (1.4) and (1.18) define $g(n)={\sum }_{k\le n}\mu (k)/k$ and the unbalanced natural prefix

$$S_n(x)=\sum _{k=1}^n\mu (k)\rho \left(\frac{1}{kx}\right).$$

Thus the approximation residual has sign $\chi +S_n$; it is not $\chi -S_n$. The coefficient cutoff passes through all positive integers $n$, and the half-line norm retains the region $x>1$.

Proposition 4.4, PDF p. 17, states that if $\zeta$ has a zero of real part $1/p$, then $S_n$ and the associated balanced sequence $V_n$ do not converge in $L^p$. In particular they do not converge in $L^2$. This last conclusion uses the known existence of zeros on the critical line, not RH. Its proof retains the tail $x>1/m$; equation (4.10) under the assumed convergence yields

$$\Vert \chi +S_m{\Vert }_p^p\ge \frac{m^{p-1}}{p-1}|g(m){|}^p.$$

The paper combines that implication with the non-little-oh statement (4.9). Proposition 4.5, PDF p. 18, also gives (4.11)–(4.12), including $\Vert \chi +S_n{\Vert }_2\ge \sqrt{n}|g(n)|$. These are quoted source results; no lower-bound scan is required to reuse them.

What this obstruction does not say

Remarks 4.6 and the paragraph following Proposition 4.5 explicitly discuss selected zero-crossing subsequences as unresolved possibilities in this source. Full-sequence nonconvergence does not by itself exclude all subsequences. A Fibonacci cutoff schedule would need a separate argument; it cannot be rejected just by changing the index name in Proposition 4.4.

Proposition 4.6, PDF p. 18, treats the distinct natural sequences $B_n$ and $F_n$. Proposition 4.7, PDF p. 19, treats fixed-coefficient series in the balanced family ${\mathcal{C}}^{nat}$ on the unit interval. Neither is silently substituted for Proposition 4.4 on the half-line.

The closure criterion permits coefficients depending on the cutoff. Its natural-dilation form is already sourced in the 2002 criterion note. The proof in that source uses damped coefficients and an order of limits; its Introduction also discusses the logarithmically mollified Selberg approximation. Those are different approximation problems from the unweighted $S_n$. Pointwise or unit-interval $L^1$ convergence of natural prefixes supplies neither half-line $L^2$ convergence nor an unconditional mollifier estimate.

Interface with actual FIB coefficients

FIB §391 uses the actual coefficients $e=\mu \ast \beta$ of §§384–386. The target transported with those coefficients is ${\Phi }_{\beta }={\sum }_d{\beta }_d{K}_d\chi$, not the unchanged $\chi$. The prefix identity retains every $d>n$ target term. The dominant-head estimate already in §384 and the existing Dirichlet-inverse budget supply an absolutely summable inverse.

That section’s additional interface is a paper application of standard dilation, convolution and convergence facts, not a new Nyman–Beurling theorem or a claim of priority. The literature obstruction is used directly. It does not settle actual FIB critical growth, a signed Robin remainder or RH.