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REVIEW 3 major objections 2 minor

An explicit formula for the discrete Laplace transform of the Möbius function yields a one-sided criterion for the Riemann hypothesis.

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2026-07-15 09:46 UTC pith:3PV6ZTT2

load-bearing objection Abstract-only: a discrete-Laplace Möbius formula with double-pole logs and a one-sided RH criterion; interesting if the residues check out, but we cannot verify them yet. the 3 major comments →

arxiv 2607.09797 v2 pith:3PV6ZTT2 submitted 2026-07-09 math.GM

Explicit formula for the discrete Laplace transform of the M\"obius function, related special functions, and a criterion for the Riemann hypothesis

Sergey Liflandsky This is my paper
classification math.GM MSC 11M2611M0611N37
keywords Riemann hypothesisMöbius functiondiscrete Laplace transformexplicit formulazeta zerosBessel functionsMertens function
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Assuming every zero of the Riemann zeta function is simple, the paper writes the discrete Laplace transform Φ(e^{-t})=∑ μ(n)e^{-nt} in closed form from the values of ζ and ζ' at the odd positive integers together with the non-trivial zeros of ζ. The formula differs structurally from the classical explicit formula for the Mertens function because the poles of the gamma factor collide with the trivial zeros of zeta, producing double poles whose residues carry an extra logarithmic term. From this representation the author extracts a criterion: the bound O(x^{-1/2}) on a suitably rescaled transform implies the Riemann hypothesis with no further assumptions, while the converse needs extra hypotheses on the zeros. The same circle of ideas produces new entire functions built from zeta that admit absolutely convergent expansions as Möbius-weighted series of Bessel functions of rotated argument.

Core claim

Under the standing hypothesis that all zeros of zeta are simple, the discrete Laplace transform of the Möbius function admits an explicit formula expressed solely in terms of ζ(2k+1), ζ'(2k+1) and the non-trivial zeros; the same formula supplies a one-sided analytic criterion for the Riemann hypothesis via the bound O(x^{-1/2}).

What carries the argument

The residue calculus at the double poles formed when poles of Γ(s) coincide with the trivial zeros of ζ(s); those residues generate the logarithmic terms that distinguish the formula from the classical Mertens explicit formula and allow the transform to be written in closed form.

Load-bearing premise

Every non-trivial zero of the Riemann zeta function is simple; if any zero has multiplicity greater than one the residue calculation that produces the stated formula changes.

What would settle it

Exhibit a multiple zero of zeta, or prove that the rescaled transform fails to be O(x^{-1/2}) while all zeros still lie on the critical line under the paper's extra hypotheses.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 2 minor

Summary. Under the standing assumption that every zero of the Riemann zeta function is simple, the paper claims an explicit formula for the discrete Laplace transform Φ(e^{-t})=∑_{n≥1} μ(n)e^{-nt}, expressed in terms of the values of ζ(s) and ζ'(s) at the odd positive integers together with the non-trivial zeros of ζ. A structural feature of the derivation is that poles of Γ(s) collide with the trivial zeros of ζ, producing double poles whose residues contain logarithmic terms; this distinguishes the formula from the classical explicit formula for the Mertens function. From the formula the authors extract a one-sided criterion for the Riemann hypothesis: the bound O(x^{-1/2}) on the transform implies RH unconditionally, while the converse requires additional (unspecified in the abstract) hypotheses on the zeros. The paper also introduces special entire functions related to ζ and asserts that they admit absolutely convergent closed forms as Möbius-weighted series of Bessel functions of rotated argument.

Significance. If the residue calculus and contour analysis are correct, the work would supply a new explicit formula for the Möbius-weighted exponential sum that is structurally different from the classical Mertens formulae, together with a clean one-sided RH criterion and closed-form expressions for a family of entire functions linked to ζ. These are potentially useful contributions to analytic number theory. The abstract already flags the simplicity assumption and the one-sided character of the RH criterion, which is a point of honesty. No machine-checked proofs, reproducible code, or fully parameter-free numerical predictions are claimed in the material under review.

major comments (3)
  1. [Abstract (explicit formula / double-pole residues)] Only the abstract is available for review. The central claim—an explicit formula obtained by residue calculus at double poles arising from the collision of poles of Γ(s) with the trivial zeros of ζ(s)—rests on analytic continuation, contour shifts, and residue bookkeeping that cannot be checked from the abstract alone. Without the full derivation, error estimates, and precise contour/shift hypotheses, the load-bearing steps of the paper remain unverifiable.
  2. [Abstract (simplicity assumption)] The standing global assumption that all zeros of ζ are simple is load-bearing for the claimed double-pole logarithmic terms. If any zero has multiplicity greater than one, the residue calculus that produces the stated formula would need re-derivation and may fail in the form announced. The abstract does not indicate how (or whether) the multiple-zero case is treated or delimited.
  3. [Abstract (RH criterion)] The RH criterion is one-sided: O(x^{-1/2}) on the transform is said to imply RH unconditionally, while the converse requires “additional hypotheses on the zeros.” Those extra hypotheses are not stated in the abstract, so the precise scope and strength of the criterion cannot be assessed. A complete manuscript must make them explicit and show they are not circular with respect to RH itself.
minor comments (2)
  1. [Abstract] The abstract is clearly written and already distinguishes the one-sided character of the RH criterion and the role of double poles; these distinctions should be preserved and expanded with precise statements of hypotheses in the full text.
  2. [Abstract (special entire functions)] Notation for the special entire functions and for the rotated-argument Bessel series should be introduced with explicit defining formulae and absolute-convergence estimates once the full text is available.

Circularity Check

0 steps flagged

No significant circularity detectable from the abstract; claims are standard residue-calculus explicit formulae, not tautological or fitted.

full rationale

Only the abstract is available. It states an explicit formula for the discrete Laplace transform Φ(e^{-t})=∑μ(n)e^{-nt} obtained under the standing assumption that all zeros of ζ are simple, expressing Φ in terms of the external objects ζ(2k+1), ζ'(2k+1) and the non-trivial zeros, via residues at double poles that arise when Γ-poles collide with trivial zeros. The RH criterion is a one-way implication (decay O(x^{-1/2}) ⇒ RH) with the converse flagged as requiring extra hypotheses; neither direction is a restatement of RH by definition. The special entire functions are introduced as having absolutely convergent Möbius–Bessel series representations. None of these steps reduces, by construction or by self-citation of an unverified uniqueness theorem, to an input that already encodes the claimed output. No fitted parameters are renamed as predictions, no ansatz is smuggled via self-citation, and no known empirical pattern is merely renamed. Residual risk is solely the abstract-only limitation (full residue calculus cannot be inspected), which does not constitute circularity under the stated rules. Score 0 is therefore the honest finding.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

Abstract-only: free parameters are not visible. The load-bearing extra axiom is simplicity of all zeta zeros. Background analytic number theory (Euler product, functional equation, Mellin transforms, residue calculus, known trivial zeros) is assumed as domain standard. The ‘special entire functions’ are introduced as new objects whose independent evidence is the claimed Bessel series, which cannot be checked here.

axioms (3)
  • ad hoc to paper All zeros of the Riemann zeta function are simple.
    Stated as a global standing assumption in the abstract; required for the residue structure of the claimed explicit formula.
  • domain assumption Standard analytic properties of ζ(s): functional equation, Euler product, known trivial zeros at negative even integers, and applicability of Mellin/Perron inversion with contour shifts.
    Implicit background for any explicit formula relating Möbius sums to zeta zeros and for the Γ–ζ pole collision mechanism.
  • ad hoc to paper Additional (unspecified in the abstract) hypotheses on the zeros needed for the converse RH implication.
    Abstract states the converse requires extra hypotheses; those hypotheses are part of the logical support for the full criterion.
invented entities (1)
  • Special entire functions related to ζ(s) with Möbius-weighted Bessel series of rotated argument no independent evidence
    purpose: Provide absolutely convergent closed forms linked to the same Möbius transform / zeta structure.
    Introduced in the abstract as new objects; independent evidence would be the claimed series identities and any growth/zero properties, which are not checkable from the abstract alone.

pith-pipeline@v1.1.0-grok45 · 6091 in / 2461 out tokens · 23376 ms · 2026-07-15T09:46:51.715572+00:00 · methodology

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read the original abstract

In this paper, we assume that all the zeros of the Riemann zeta function are simple. Under this assumption we give an explicit formula for the function $\Phi(e^{-t})=\sum_{n=1}^{\infty}\mu(n)e^{-nt}$, as a function of the values of $\zeta(s)$ and $\zeta'(s)$ at the odd integers and as a function of the zeros of $\zeta(s)$. A structural feature distinguishes this formula from the classical explicit formula for the Mertens function: the poles of $\Gamma(s)$ collide with the trivial zeros of $\zeta(s)$, producing double poles whose residues contain a logarithmic term. Using this formula, we give a criterion for the Riemann hypothesis: the bound $O(x^{-1/2})$ on the transform implies the Riemann hypothesis unconditionally, while the converse direction requires additional hypotheses on the zeros. We also introduce special entire functions related to $\zeta(s)$ and show that they admit absolutely convergent closed forms as M\"obius-weighted series of Bessel functions of rotated argument.

Figures

Figures reproduced from arXiv: 2607.09797 by Sergey Liflandsky.

Figure 1
Figure 1. Figure 1: The positively oriented contour RN = C1 ∪ C2 ∪ C3 ∪ C4 of (5.15): right edge C1 (blue) on σ = σ0, top edge C2 (red) at t = TN , left edge C3 (green) on σ = 1 2 − 2N, bottom edge C4 (orange) at t = −TN ; the heights ±Tν of Lemma 5.2 are marked on the imaginary axis. Crosses: nontrivial zeros of ζ, drawn on the critical line (shaded). Dots: poles of Γ. Circled dots: collisions of the poles of Γ with the triv… view at source ↗

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