bibkey: liflandsky2026mobiuslaplace authors: Sergey Liflandsky year: 2026 title: Explicit formulas for the Möbius Laplace transform and a criterion for the Riemann hypothesis and simple zeros doi: null url: https://arxiv.org/abs/2607.09797v4 claim: The exact Möbius Laplace endpoint implies RH and simple zeros; the converse retains a uniform Fejér zero-sum bound, and the endpoint implication already specializes Báez-Duarte’s convolution theorem. strata_touched: [] license: citation-only triage: anchor

The Möbius Laplace endpoint and its additional obligations

The primary version is arXiv:2607.09797v4, revised 14 September 2026. The title above is the title in its v4 full text; the abstract landing page retains a different title and abstract. The inspected scope is Theorem 1.3, §7.1’s hypothesis and Theorem 7.2, §8.1’s Propositions 8.2–8.3 and Remark 8.4, and §8.2’s explicit predecessor correspondence. The complete contour, special-function and Fejér proofs are not independently audited. This is a preprint source note, with no Lean verification or original FIB criterion claimed.

Endpoint implication and converse conditions

Write

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

Theorem 1.3(a) states that

$${\Phi }_{\mu }(t)=O(t^{-1/2})(t\downarrow 0)⟹\text{RH and simplicity of every nontrivial zero}.$$

The implication does not assume RH or simplicity. It is stronger than a criterion whose conclusion is RH alone.

Under RH and simplicity, Theorems 1.3(b) and 7.2 make this endpoint equivalent to the existence of one constant $C_0$ such that, simultaneously for every $T>0$ and $t>0$,

$$\left|\sum _{|\gamma |<T}\left(1-\frac{|\gamma |}{T}\right)\frac{\Gamma (1/2+i\gamma )}{{\zeta }^{\prime }(1/2+i\gamma )}{t}^{-i\gamma }\right|\le C_0.$$

Both positive and negative ordinates are retained. Absolute summability of these residue coefficients is a sufficient alternative condition; no linear-independence hypothesis on the ordinates is required. The source explicitly does not derive the uniform bound from RH and simplicity alone. Pointwise convergence of a smoothed zero sum does not discharge this two-parameter uniform estimate.

Proposition 8.2 states that an endpoint with logarithmic loss $O(t^{-1/2}(\log (1/t){)}^a)$, $a\ge 0$, implies RH and bounds every nontrivial zero’s multiplicity by $a+1$. Proposition 8.3 states that the exact endpoint also forces square-summability of the residues. Remark 8.4 distinguishes that necessary condition from the absolute summability used as a sufficient hypothesis; it does not prove the latter.

The endpoint implication already has a predecessor

Section 8.2 explicitly specializes Báez-Duarte’s Theorem 4.1. In that paper’s multiplicative-convolution notation, take

$$g(v)=\sum _{n\le v}\frac{\mu (n)}{n},h(u)=u^{-1}{e}^{-1/u},\varphi (u)=u{h}^{\prime }(u).$$

The source gives the exact parameter map

$$G_{\varphi }(y)=y^{-1}{\Phi }_{\mu }(1/y),{\varphi }^{\wedge }(s)=s\Gamma (s+1).$$

Thus the small-$t$ bound becomes $G_{\varphi }(y)=O(y^{-1/2})$ as $y\to \infty$. The source verifies the required Mellin convergence, extension and nonvanishing conditions for this test. The endpoint implication is therefore an existing convolution criterion, rather than a new theorem to reproduce in FIB coordinates.

Consequence for the actual FIB dilation filter

The existing smoothing note already records the actual two-way dilation identities for $e=\mu \ast \beta$ and the summable inverse $\gamma ={\beta }^{-1}$. Applying its existing dilation-and-triangle argument to the weighted supremum ${\sup }_{t>0}\sqrt{t}|\Phi (t)|$ uses weights $d^{-1/2}$. The existing inverse budgets make both transport constants finite.

An exact FIB endpoint would consequently carry the source’s stronger RH-and-simplicity obligation, together with the corresponding uniform zero-sum condition. No endpoint bound for the actual FIB or Möbius transform is supplied by the filter identities, the five-pattern geometry, or this note. The RH-only $L^p$ criterion for every $1\le p<2$ in the existing smoothing note remains a different target. Neither criterion supplies the outstanding complete signed Robin estimate.