bibkey: molnar2026reciprocalrobin authors: Grant Molnar year: 2026 title: Reciprocal LCM powers and Robin’s criterion doi: null url: https://arxiv.org/abs/2610.12816v1 claim: “The reciprocal LCM-power function is the existing increment source b_kappa. The preprint reports an RH-equivalent unbounded-parameter criterion and unconditional normalized inequalities for all n > 5040 at 1 < kappa <= 100000; a bounded parameter interval does not settle classical Robin.” strata_touched: [] license: citation-only triage: anchor

Reciprocal LCM powers: reuse the existing source and parameter criterion

The inspected primary is arXiv:2610.12816v1, submitted 8 October 2026, 23 pages. The definitions, theorem statements, parameter-criterion argument and certification description cited below were read in that version. The full analytic proofs and computer certificates have not been independently verified or Lean compiled here. These results belong to Molnar; this note adds no originality or peer-review claim.

The reciprocal object already occurs in the project

With $Z(n)=\sigma (n)/n$, Definition 1.2 and equations (5)–(7), printed pp.1–2, give

$${\sigma }_{-1}^{[\kappa ]}(n)=\sum _{d\mid n}\mu (d)Z(n/d{)}^{\kappa }=b_{\kappa }(n),Z(n{)}^{\kappa }=\sum _{d\mid n}{b}_{\kappa }(d).$$

This is the same increment source identified in the Fan–Kobayashi–Molnar note. Its local factor is $Z(p^a{)}^{\kappa }-Z(p^{a-1}{)}^{\kappa }$. For integral $k$, equation (8), printed p.2, already supplies

$$b_k(n)=\sum _{\mathrm{lcm}(d_1,\dots ,d_k)=n}\frac{1}{d_1\cdots d_k}.$$

Reproving this expansion or introducing the same source under a FIB name would duplicate existing work. The distinct nonreciprocal function ${\sigma }^{[\kappa ]}$ in the earlier paper must not be substituted for it.

Theorem 2.8, printed pp.6–8, supplies complete Selberg–Delange expansions for both ${\sum }_{n\le x}n{b}_{\kappa }(n)$ and ${\sum }_{n\le x}{b}_{\kappa }(n)$ at each fixed real $\kappa >1$. This is a directly citable weighted expansion for the corrected Balakrishnan–Pétermann input discussed in the earlier note. Its $\kappa$-dependent errors do not pay the project’s required growing-parameter error at $s=y\log y$.

Normalize before applying the Robin criterion

Definition 1.4, printed p.2, uses

$${\hat{\rho }}^{[\kappa ]}(n)={\left(\frac{n{b}_{\kappa }(n)}{{\kappa }^{\omega (n)}}\right)}^{1/(\kappa -1)},\kappa >1,$$

where $\omega (n)$ counts distinct prime divisors. This normalization is part of the tested function, not an optional factor in its budget. Lemma 3.1 and equation (53), printed p.13, state that for every fixed integer $n>1$,

$${\hat{\rho }}^{[\kappa ]}(n)<Z(n),{\hat{\rho }}^{[\kappa ]}(n)\nearrow Z(n)(\kappa \to \infty ).$$

Theorem 3.4, printed p.14, states that for any fixed unbounded set $\mathcal{K}\subset (1,\infty )$, RH is equivalent to

$${\hat{\rho }}^{[\kappa ]}(n)<e^{\gamma }\log \log n\text{for every integer }n>5040\text{ and every }\kappa \in \mathcal{K}.$$

The same theorem allows a common eventual integer cutoff for all $\kappa \in \mathcal{K}$. Its reverse implication uses Robin’s strict violations under RH failure; passing a strict inequality to a limit alone would yield only a weak inequality and is not its proof.

The bounded-parameter theorem is unconditional

Theorem 4.7, printed pp.18–19, reports an explicit finite cutoff $N_{{\kappa }_0}$ for every real ${\kappa }_0>1$, beyond which the normalized inequality holds for every real $1<\kappa \le {\kappa }_0$. Theorem 4.9, printed p.20, reports the stronger complete range

$${\hat{\rho }}^{[\kappa ]}(n)<e^{\gamma }\log \log n(n>5040,1<\kappa \le 100000).$$

This is an all-integer statement for a bounded interval of the reciprocal parameter. It is not a new finite verification of the classical function $Z(n)$, nor a result for unbounded $\kappa$. The source combines an analytic endpoint-transfer theorem with directed interval certificates; its code and certification section, printed p.22, names the MIT-licensed reciprocal-robin repository. Those certificates have not been rerun here.

Corollary 4.8, printed p.19, already states that along any sequence of strict classical Robin violations tending to infinity, the reciprocal parameter at which each violation becomes visible must tend to infinity. Thus increasing a finite parameter ceiling alone does not exclude such a sequence.

Consequences for the FIB and same-source Robin routes

The parameter $k$ in the LCM expansion counts divisor entries; the depth of a [null,2,3,2 5,5] address counts successive Zeckendorf windows. They are different parameters. The source supplies no theorem identifying either parameter with the other or with a Fourier cutoff. Its normalized extremizers in section 4 likewise optimize a different function from the classical strict CA stock. Appendix A, printed pp.21–22, gives an explicit obstruction to using unnormalized reciprocal endpoints for the normalized Robin test.

Reuse the source’s LCM expansion, fixed-parameter expansions, criterion and certified bounded range before seeking a new estimate. For the selected-source route, the signed $I_{\psi }(A)$ lower bound remains unpaid; for the increment-moment route, the growing-$s$ error remains unpaid. This preprint supplies neither missing estimate, and its bounded result cannot be presented as a proof of classical Robin or RH.