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

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Pith's one-line read A decay bound on the discrete Laplace transform of the Möbius function forces the Riemann hypothesis, and a new explicit formula exposes the mechanism.

desk verdict A sound, genuinely new explicit formula for the Laplace-kernel Möbius sum, plus a clean unconditional proof of one implication of an RH criterion; the main caveat is that the headline identity itself depends on an unproved simplicity hypothesis. read the letter →

arxiv 2607.09797 v3 pith:3PV6ZTT2 submitted 2026-07-09 math.GM

Sergey Liflandsky This is my paper · ORCID
classification math.GM MSC 11M0611M2633C10
keywords RiemannzetafunctionMöbiushypothesisexplicitformulaBesselfunctionssimplezerosMellintransformdiscreteLaplace
topics The Riemann Hypothesis
open problems The Riemann Hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central aim is a criterion for the Riemann hypothesis from the discrete Laplace transform of the Möbius function, Φ(e^{-x}) = Σ μ(n)e^{-nx}. It proves that if Φ(e^{-x}) = O(x^{-1/2}) as x→0+, then every nontrivial zero of ζ lies on the critical line, with no hypothesis on the zeros needed for that direction. The same machinery, under the assumption that all nontrivial zeros are simple, yields an explicit formula for Φ(e^{-x}) as a zero sum plus entire functions, in which the collision of Gamma poles with trivial zeta zeros creates double poles and a logarithmic residue not present in the classical Mertens formula. Three of the resulting entire functions admit absolutely convergent closed forms as Möbius-weighted Bessel series. A sympathetic reader would care because the bound-to-RH direction turns a single asymptotic estimate into the full Riemann hypothesis.

What carries the argument

The central object is the Mellin inversion formula Φ(e^{-x}) = (1/2πi)∫_{3/2-i∞}^{3/2+i∞} Γ(s)/ζ(s) x^{-s} ds. Shifting this contour left past the trivial zeros is the mechanism: residues at odd negative integers produce κ, residues at even negative integers are evaluated as double poles whose log term yields β, λ, and υ, and residues at nontrivial zeros give the ζ'(ρ)-weighted zero sum. Two estimates carry the contour shift: a uniform lower bound on |Γ(σ+it)| supplying exponential cancellation on the moving line Re(s)=1/2−2N, and a classical selection of horizontal heights avoiding zeros of ζ. For the Riemann-hypothesis criterion, the key identity is ζ(s)G(s)=Γ(s), extended from Re(s)>1 to

What would settle it

Evaluate both sides of the explicit formula at x=0.1, x=0.5, and x=1 to 60 significant digits; any mismatch between the direct Möbius sum and the zero-plus-special-function side would falsify the residue computation. For the RH implication there is no numerical experiment that could directly falsify it—a counterexample would require RH to be false and the O(x^{-1/2}) bound to hold.

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Extended reading notes

Core claim

Under the assumption that every nontrivial zero of ζ is simple, the paper proves the exact identity Φ(e^{-x}) = Σ_ρ Γ(ρ)x^{-ρ}/ζ'(ρ) + πκ(x) + 2λ(x) + 4υ(x) − 2β(x)log(2πx) − 2. The distinguishing structural feature is that Γ(s)x^{-s}/ζ(s) has double poles at the negative even integers, where poles of Γ collide with trivial zeros of ζ; their residues contain the logarithmic term and the digamma and logarithmic-derivative weights of ζ. The paper also proves an unconditional implication: if Φ(e^{-x})=O(x^{-1/2}) near 0, then its Mellin transform G(s) is holomorphic for Re(s)>1/2, and the identity theorem applied to ζ(s)G(s)−Γ(s) rules out zeros with Re(s)>1/2; the functional equation then rule

Load-bearing premise

The load-bearing premise is the O(x^{-1/2}) bound on the Möbius Laplace sum near 0; it is an unproved regularity condition, and if it fails, the unconditional implication has nothing to act on (while the explicit formula separately leans on the unproved simplicity of all nontrivial zeros).

Editorial extensions

If this is right

  • A proof of the O(x^{-1/2}) bound would immediately settle the Riemann hypothesis, with no separate control of the zeros.
  • The explicit formula determines the Möbius Laplace transform exactly from the zeros, through Γ(ρ)x^{-ρ}/ζ'(ρ), plus four computable entire functions and a logarithmic term.
  • The Möbius–Bessel identities give absolutely convergent closed forms for κ, β, and λ as series of J0 with rotated argument, avoiding conditionally convergent sums entirely.
  • Under RH, simplicity, and the absolute-convergence hypothesis, the criterion is two-way: the x^{-1/2} decay holds if and only if RH holds.
  • If simplicity is dropped, the formula survives with multiplicity-adjusted derivative residues, so the structural mechanism does not depend on Hypothesis (S).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The ζ(s)G(s)−Γ(s) identity-theorem trick never divides by ζ near a possible zero, so it may convert other decay bounds on Möbius-type averages into zero-free regions for related Dirichlet series.
  • The paper's closing remark on the e^{-n^α x} family suggests a testable dichotomy: logarithmic residues should appear exactly for rational α and not for irrational α, since Gamma poles and trivial zeta zeros collide only then.
  • The converse's extra hypotheses indicate the O(x^{-1/2}) condition is far stronger than RH itself, so the criterion is best read as a sufficient condition; it is unlikely to offer an easy path to RH through the converse direction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies the function Phi(e^{-x}) = sum_{n=1}^infty mu(n) e^{-nx}, whose Mellin transform is Gamma(s)/zeta(s). Under the assumption that all nontrivial zeros of zeta are simple (Hypothesis S), it derives an explicit formula expressing Phi(e^{-x}) as a sum over nontrivial zeros plus four explicitly defined entire functions kappa, beta, lambda, upsilon, with a logarithmic term -2 beta(x) log(2 pi x) that arises from the collision of poles of Gamma with the trivial zeros of zeta. It also proves absolutely convergent Mobius-Bessel series representations for kappa, beta, lambda. The central criterion is Theorem 1.3(a): if Phi(e^{-x}) = O(x^{-1/2}) as x -> 0+, then the Riemann hypothesis follows, with no hypothesis on the zeros of zeta. The converse direction, Theorem 1.3(b), is proved under RH together with Hypotheses S and H.

Significance. If correct, the explicit formula is a new structural result, distinguished from the classical Mertens explicit formula by the presence of double poles at the trivial zeros and the resulting logarithmic term. The unconditional direction of the RH criterion is a new sufficient condition of the Hardy-Littlewood-Riesz type. The paper's strengths are the detailed and transparent proof structure, the cross-checked residue computations in Sections 3-4 and Appendices D-H, and the non-circular identity-theorem argument in Theorem 1.3(a), which applies the identity theorem to zeta*G - Gamma rather than to 1/zeta. The main limitations - that the explicit formula is conditional on unproved Hypothesis S and the converse direction on unproved Hypothesis H - are explicitly disclosed, and Remark 5.5 gives an unconditional form of the explicit formula with multiplicity-adjusted zero terms. I found no load-bearing technical error in the proof of the central implication.

minor comments (6)
  1. [Abstract] The phrase "all the zeros of the Riemann zeta function are simple" should be qualified as "all nontrivial zeros", matching Hypothesis (S) in Section 1.1. The trivial zeros are simple and are not at issue in the conjecture.
  2. [Section 5.2, Lemma 5.2] The lemma states the lower bound (5.14) for -1 <= sigma <= 2 and cites Titchmarsh Theorem 9.7. Since the standard theorem is often stated for 1/2 <= sigma <= 2, please add one sentence explaining how the extension to -1 <= sigma <= 1/2 follows from the functional equation, or confirm that the cited theorem already contains this range.
  3. [Section 7] The remark that the first zero has imaginary part 14.134... is not used in the displayed pairing argument (7.3)-(7.9). It can be removed or explicitly connected to the claim that rho and 1-rho lie in opposite halves of the strip.
  4. [Acknowledgements] The statement that the main identity was "verified numerically to thirty significant digits at several values of x" is not reproducible without the x-values and the computed values. Either supply the numerical data or soften the claim.
  5. [Figure 1] The labels T_N-1 and T_N-2 on the imaginary axis are unexplained; since the heights are supplied by Lemma 5.2, clarify that these are schematic markers, not all distinct ordinates used in the proof.
  6. [Section 5.3, Eq. (5.19)] In the displayed bound, the constant C_2 absorbs the factor (2 pi x)^{-1/2} from (2 pi x)^{2N-1/2}. It would be clearer to define C_2 accordingly, or to keep (2 pi x)^{2N} and note the absorbed factor explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RH criterion (Theorem 1.3(a)) is an identity-theorem argument from the assumed bound, and unproved S/H affect only explicitly disclosed auxiliary directions.

full rationale

The central derivation is self-contained and not circular. Theorem 1.3(a) assumes only the O(x^{-1/2}) bound on Phi(e^{-x}) and uses it, via Lemma 8.1, to construct G(s) holomorphic on Re(s)>1/2. The function F(s)=zeta(s)G(s)-Gamma(s) is holomorphic on the connected domain D={Re(s)>1/2}\{1}, because it contains zeta(s) itself rather than 1/zeta(s); it vanishes on Re(s)>1 by the previously proved identity (1.5), so the identity theorem forces zeta(s)G(s)=Gamma(s) on all of D. Evaluating at a hypothetical zero rho with Re(rho)>1/2 gives Gamma(rho)=0, impossible, and the functional equation excludes zeros with Re(s)<1/2. This never assumes holomorphy of 1/zeta in Re(s)>1/2 and uses neither Hypothesis (S) nor Hypothesis (H). The explicit formula (1.19) is derived by a standard contour shift and residue computation, with Hypothesis (S) used only to write the zero residues in closed form (2.1); Remark 5.5 explicitly states the unconditional multiplicity-adjusted version. The converse direction (Theorem 1.3(b)) is an implication, not an input: it uses RH, S, and H to derive the bound. The Möbius-Bessel identities of Theorem 1.2 are direct algebraic rearrangements of the definitions (1.17)-(1.18) via the Dirichlet series for 1/zeta, with absolute-convergence interchanges supplied; they are auxiliary and are not used in the RH criterion. There are no fitted parameters relabeled as predictions and no load-bearing self-citation chain; all external citations are to standard results (Titchmarsh, DLMF, etc.). The paper's genuine fragilities, namely the unproved O(x^{-1/2}) premise and the auxiliary Hypotheses S and H, are clearly disclosed as assumptions rather than disguised as conclusions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted numbers or ad hoc constants appear; the central formulas introduce no free parameters. Hypotheses S and H are explicit unproved conditions. The four special functions κ, β, λ, υ are new definitions but are ordinary entire functions, not physically invented entities.

assumptions (5)
  • domain assumption All nontrivial zeros of ζ are simple (Hypothesis S).
    Used to evaluate residues at the zeros as Γ(ρ)x^{-ρ}/ζ'(ρ) in §2 (2.1); if a multiple zero exists, (1.19) must be replaced by derivative terms (Remark 5.5).
  • domain assumption Absolute convergence of Σ |Γ(ρ)|/|ζ'(ρ)| over nontrivial zeros (Hypothesis H).
    Introduced in §7 and used in the converse direction Theorem 1.3(b); the paper notes S does not imply H.
  • standard math Titchmarsh ordinate-selection lemma provides heights Tν with |ζ(σ±iTν)|^{-1} ≤ Tν^{A0} for -1≤σ≤2.
    Used in horizontal-contour estimates (5.14); cited from [19, Theorem 9.7], not proved in the paper.
  • standard math Mellin inversion lemma (Appendix A) via Fourier inversion for functions of locally bounded variation.
    Establishes the starting inversion (1.6).
  • standard math Reflection formula and functional equation of ζ.
    Used throughout Sections 3-5 and the appendices to evaluate residues and derive the reflected representation.

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Pith. "Pith review of Explicit formula for the discrete Laplace transform of the M\"obius function, related special functions, and a criterion for the Riemann hypothesis." pith.science (2026). https://pith.science/paper/3PV6ZTT2

@misc{pith2026260709797,
  author       = {Pith},
  title        = {Pith review of: Explicit formula for the discrete Laplace transform of the M\"obius function, related special functions, and a criterion for the Riemann hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3PV6ZTT2}},
  note         = {Machine review of arXiv:2607.09797}
}
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 the authors.

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 trivial zeros … view at source ↗
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), 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) for illustration only; Theorem 1.1 does not assume the Riemann hypothesis, and Hypothesis (S) does no… view at source ↗

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