bibkey: baezduarte2005mobiusconvolutions authors: “Luis Báez-Duarte” year: 2005 title: “Moebius-convolutions and the Riemann hypothesis” doi: null url: “https://arxiv.org/abs/math/0504402v1” claim: “Section 2 records classical weighted Möbius growth and Littlewood’s RH criterion; the Fibonacci harmonic weights use these existing tools without rederiving the criterion.” strata_touched: [] license: citation-only triage: anchor

Weighted Möbius sums and convolution

The primary version is arXiv math/0504402v1, published 2005-04-20. The manuscript identifies its text as 2004-11-22, with comments added 2005-04-19. The arXiv record supplies no journal reference or DOI. The inspected original TeX scope is the abstract, Sections 1–2 and the hypotheses and conclusion of the GL1 lemma in Section 3.1; the complete analytic proofs and the other results are not independently certified here. No source text is vendored. The source archive has SHA256 3654ec2a72bdae5d567ab0968d7283a636894928c07464a7986b19bcc491aeef.

In Section 2, write

[ M(x)=\sum_{n\le x}\mu(n),\qquad g_\mu(x)=\sum_{n\le x}\frac{\mu(n)}n. ]

The source’s equation labeled growthMandg states, for (\alpha\in[1/2,1)),

[ M(x)\ll x^\alpha\quad\Longleftrightarrow\quad g_\mu(x)\ll x^{\alpha-1}. ]

The same section records Littlewood’s classical criterion (\mathrm{RH}\Longleftrightarrow g_\mu(x)\ll_\varepsilon x^{-1/2+\varepsilon}) for every positive (\varepsilon). Its equation labeled growthg uses the unconditional bound (M(x)\ll x(\log x)^{-3}) to obtain (g_\mu(x)\ll(\log x)^{-2}). These are literature inputs, not new Fibonacci results or a Lean formalization.

For the actual Fibonacci kernel (e=\mu*\beta), finite convolution gives

[ g_e(D)=\sum_{m\le D}\frac{e_m}{m} =\sum_{d\le D}\frac{\beta_d}{d},g_\mu(D/d). ]

The FIB volume §394 uses this classical relation and the existing actual-input zero moment. Its claim that every finite actual (g_e(D)) is nonzero depends separately on the integer Fibonacci atoms, the unit correction, and the golden norm. That actual-source conclusion and the cofactor prime-panel asymptotic are repository derivations, not statements attributed to this paper. The paper’s symbol (\phi) for a convolution test function is not the FIB volume’s golden ratio (\varphi).

The existing Mellin convolution input

Section 2 defines

$$N_a(f)={\int }_0^{\infty }|f(t)|t^{-a-1}dt.$$

Section 3.1, the lemma labeled GL1, assumes $N_0(\varphi )<\infty$ and uses the classical convolution algebra on the multiplicative group $(0,\infty )$ with Haar measure $dt/t$. Its norm conclusion is

$$N_0(G\varphi )\le N_0(g)N_0(\varphi ),g(x)=\sum _{n\le x}\frac{\mu (n)}{n}.$$

This is an existing absolute-convolution tool. The project’s finite $N_{\alpha }(R)$ for $0<\alpha <1$ does not by itself supply $N_0(R)$; the displayed $O(1+\log y)$ high-quotient envelope alone also does not prove that latter integral finite. No substitution $\varphi =R$ or identification $G\varphi =I_{\psi }$ is asserted from those bounds.

The actual discrete adjacent-dilation difference in FIB §397 instead retains the jump intervals of $K$ and the low-quotient continuous part of $R$. It verifies the countable integral-norm premise for that same-source series before using the existing integral-sum theorem. The classical norm theorem and its Mellin formulation are not new FIB results; the manuscript application remains without a complete Lean verification or an RH/Robin conclusion.

The exact endpoint already has a simplicity consequence

The additional inspected original TeX scope is Theorem 4.1, labeled rhs, and its stated hypotheses and conclusion. In the paper’s notation, RHS means RH together with simple nontrivial zeros. The theorem assumes a Mellin-proper test $\varphi$, an extension ${\varphi }^{\wedge }\in A_c[-1/2,1]$, and nonvanishing on the boundary line $\Re s=-1/2$. It concludes

$$G_{\varphi }(y)=O(y^{-1/2})(y\to \infty )⟹\text{RH and simple nontrivial zeros}.$$

The Liflandsky v4 source note records its §8.2 specialization with $h(u)=u^{-1}{e}^{-1/u}$ and $\varphi (u)=u{h}^{\prime }(u)$: $G_{\varphi }(y)=y^{-1}{\Phi }_{\mu }(1/y)$ and ${\varphi }^{\wedge }(s)=s\Gamma (s+1)$. That source supplies the test’s required Mellin hypotheses. The original theorem’s complete proof is not independently certified here. The original proof writes $\varphi (s)$ in its nonvanishing step, then $x^{\sigma -3/2}$ and $1/(\sigma -1/2)$ in its final estimate while taking $\sigma \downarrow -1/2$. These notation and sign issues do not match that stated boundary limit. The theorem statement is cited as written; no silent correction or certification of that argument is claimed.

This exact endpoint has a stronger conclusion than the $y^{-1/2+\epsilon }$ RH criterion with a loss for every $\epsilon >0$. Removing that loss is an additional mathematical obligation; it is not achieved by renaming the existing FIB convolution or dilation filter.