bibkey: verjovsky2026mobiussmoothing authors: Alberto Verjovsky year: 2026 title: “How Random Is the Möbius Function? Smoothing, Probability, and the Riemann Hypothesis” doi: null url: https://arxiv.org/abs/2607.25002v2 claim: “Theorem 3.1 states the Möbius Laplace Lp criterion for every 1≤p<2; Proposition 2.2 gives the zero-free half-plane from one exponent. The actual FIB multiplier is an invertible dilation filter, not an unconditional membership proof.” strata_touched: [] license: citation-only triage: anchor

The existing smoothed Möbius criterion and its FIB filter

The primary version is arXiv:2607.25002v2, submitted 14 August 2026, 28 pages. The inspected original scope is Sections 2–3: Propositions 2.1–2.2, Theorem 3.1 and Remarks 3.2–3.3. The selected proofs were read; the rest of the manuscript and its probabilistic and dynamical interpretations are not independently audited. No Lean verification or original FIB criterion is asserted.

For

$${\Phi }_{\mu }(t)=\sum _{n\ge 1}\mu (n)e^{-nt},t>0,$$

Proposition 2.1 gives the classical Mellin transform ${\int }_0^{\infty }{\Phi }_{\mu }(t)t^{s-1}dt=\Gamma (s)/\zeta (s)$ for $\Re s>1$. Proposition 2.2 states that one $L^p(0,\infty )$ bound, $1<p<\infty$, implies $\zeta (s)\ne 0$ on $\Re s>1/p$. Theorem 3.1 states

$$\mathrm{RH}⟺{\Phi }_{\mu }\in L^p(0,\infty )\text{for every }1\le p<2.$$

Remark 3.2 explicitly excludes an endpoint assertion at $p=2$. A fixed exponent below two does not give the full criterion, and a random coefficient statement is not a statement about the deterministic Möbius sequence.

The actual FIB coefficients satisfy $e=\mu \ast \beta$, with ${\beta }_n=\log (1-(-{\phi }^{-2}{)}^n)$ and the already established absolutely summable inverse $\gamma ={\beta }^{-1}$. Define ${\Phi }_e(t)={\sum }_{n\ge 1}{e}_n{e}^{-nt}$. Absolute convergence at every $t>0$ and the existing convolution identities give the two dilation formulas

$${\Phi }_e(t)=\sum _{d\ge 1}{\beta }_d{\Phi }_{\mu }(dt),{\Phi }_{\mu }(t)=\sum _{d\ge 1}{\gamma }_d{\Phi }_e(dt).$$

These are applications of the established filter, not new analytic criteria. The standard dilation norm $\Vert g(d\cdot ){\Vert }_p=d^{-1/p}\Vert g{\Vert }_p$ and the triangle inequality give, whenever the right side is finite,

$$\Vert {\Phi }_e{\Vert }_p\le \left(\sum _d|{\beta }_d|d^{-1/p}\right)\Vert {\Phi }_{\mu }{\Vert }_p,\Vert {\Phi }_{\mu }{\Vert }_p\le \left(\sum _d|{\gamma }_d|d^{-1/p}\right)\Vert {\Phi }_e{\Vert }_p.$$

Both filter constants are finite by the existing FIB inverse budget. Thus the source supplies a reusable alternative test for the same actual arithmetic input. It does not supply the still missing $L^p$ membership for all exponents below two, a signed Robin bound, or a geometric rule that forces either. This adaptation has no new retained Lean wrapper.