Jacob's ladders and new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theory based on sums of elementary $\zeta$-pulses

Jan Moser

REVIEW 4 major objections 6 minor 9 references

Jacob's ladders and new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theory based on sums of elementary $\zeta$-pulses

T0 review · 4 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that the sum of zeta-zero gap areas over a Jacob's-ladder interval equals (1−c)T+O(√T), and turns that identity into a new ζ-condition—never equal to 1 on Fermat rationals—which it calls a new ζ-equivalent of the Fermat-Wi

desk verdict A derivational exercise that repackages the author's earlier ζ-functional; the 'Fermat-Wiles equivalents' are tautologies, and the one new lemma rests on an unproved zero-gap estimate. read the letter →

arxiv 2607.17731 v1 pith:SNWNMSMY submitted 2026-07-20 math.NT

Jan Moser This is my paper · ORCID

classification math.NT MSC 11M0611M26

keywords Jacob'sladdersRiemannzetafunctioncriticallineHardy-Littlewoodintegralzeta-functionalsFermat-WilestheoremBonnetmeanvaluehypothesis

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 aims to convert the classical integral of |ζ(1/2+it)|² on the critical line into an exact-looking sum over the gaps between consecutive zeros of Z(t). Its central lemma states that the sum of zero-to-zero area integrals over the interval (T,¹T), where ¹T is the first reverse iteration of Jacob's ladder, equals (1−c)T+O(√T). Substituting T=x/(1−c)τ makes this sum a ζ-functional with value x; for Fermat rationals x=(x^m+y^m)/z^m the value is never 1, giving the claimed ζ-equivalent of the Fermat-Wiles theorem. Under the Riemann hypothesis, each zero-gap area is represented as an isolated rectangle via the Bonnet mean-value theorem, so the same functional and Fermat-equivalent hold for sums of rectangular ζ-pulses. This matters because it recasts a continuous integral identity as a discrete sum over zero gaps—a more structural object—and ties an arithmetic statement to the distribution of zeta zeros.

What carries the argument

Jacob's ladder φ₁(T) and its reverse iterations define a special interval (T,¹T) on which the integral of Z² is almost exactly (1−c)T. The interval serves as a counting frame: the zeros γ_n lying inside partition the integral into zero-to-zero curvilinear areas A₀(n). The Bonnet mean-value theorem, applied under the Riemann hypothesis's unimodality of Z² between zeros, converts each curvilinear area into a rectangle B(n) of equal area, so the same asymptotic identity holds for sums of elementary rectangular pulses.

What would settle it

Evaluate the sum Σ_{T<γ_n<¹T} ∫_{γ_n}^{γ_{n+1}} Z² for a range of large T using numerical values of the zeros and Z; if the difference from (1−c)T grows faster than the claimed O(√T)—or if the edge gaps |γ₁−T| and |¹T−γ_{N+1}| exceed T^{1/6}—the central identity is false.

Watch

Extended reading notes

Core claim

Lemma 1 is the load-bearing identity: for T→∞, Σ_{T<γ_n<¹T} ∫_{γ_n}^{γ_{n+1}} Z(t)² dt = (1−c)T + O(√T), where ¹T=φ₁^{-1}(T) is the first reverse iteration of Jacob's ladder and γ_n are the zeros of Z(t) on the critical line. The paper derives it by partitioning the integral over (T,¹T) into integrals over consecutive zero-to-zero intervals, using an elementary bound on the edge gaps. Substituting T=x/(1−c)τ yields a ζ-functional whose value at every fixed x>0 is x, and for Fermat rationals x=(x^m+y^m)/z^m the value is never 1. On the Riemann hypothesis, the unimodality of Z² between zeros lets the Bonnet mean-value theorem replace each zero-gap area by the area of a rectangle B(n)=Z²(t0(n))

Load-bearing premise

The argument relies on the unproved O(T^{1/6}) bound on the distance from T and ¹T to the nearest zero of Z(t) on the critical line; if the edge gaps are larger, the edge integrals could dominate the main term and Lemma 1 would fail.

Editorial extensions

If this is right

The identity (3.12) yields a discrete sum-form ζ-functional (4.2) that equals x for every fixed x>0, matching the original integral functional.
For Fermat rationals, the ζ-condition (4.5) says the normalized zero-gap sum never approaches 1; a solution to x^m+y^m=z^m would force the limit to be exactly 1, so the condition reproduces the Fermat-Wiles theorem.
On the Riemann hypothesis, the same functional and the same Fermat-equivalent hold for sums of rectangle-shaped pulses B(n), showing that isolated zeta-pulses carry the same arithmetic information as the continuous zero-gap areas.
The ratio identity (6.2) gives a conservation law linking the maximum and mean values of Z² on each zero gap to the gap width and the Bonnet-point width.

Reading between the lines

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

The construction suggests a numerical test: for finite τ, compute (1/τ)Σ over zeros up to [x/(1−c)τ] and compare with x; the residuals would reveal the true size of the O(√T) error term and whether the zero-gap discretization behaves as claimed.
The same 'sum over zero gaps' idea could be applied to other zeta-functionals—averages of powers of |ζ(1/2+it)|, for instance—producing a family of discrete pulse codes with arithmetic meaning.
Because the Fermat condition only relies on no Fermat rational equaling 1, the ζ-functional can in principle act as a detector for any family of rational numbers that encodes solutions to a Diophantine equation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

You should know this paper is exactly what it says: a reshuffling of the author's own functional. The sum-variants follow from the almost linear formula (3.1) and the functional (4.3) by splitting the integral at the points γ_n. There is no new arithmetic input, and the 'ζ-equivalents' are logically equivalent to the statement that the limit equals the Fermat rational; since a counterexample to FLT would make that rational 1, the whole theorem is just a tautology. That is not a proof of FLT, and the author does not claim it is.

What is good: the paper is transparent about its structure. It says these are equivalents, not proofs, and the manipulations are straightforward once you accept (3.1). The rectangle-pulse rewriting under RH (Section 5) is a correct application of the Bonnet mean-value theorem conditional on RH. The notation is consistent, and the paper does not hide behind undefined terms except for the Jacob's-ladder apparatus, which is self-consistent.

The soft spots are real. The first is (3.10): an unproved assertion that the distance from T to the first and last zero of Z(t) on the relevant interval is O(T^{1/6}). No proof or reference is given. Known results give at best O(T^{1/6+ε}) for the existence of a nearby zero, which would turn the error in (3.11)–(3.12) into O(T^{1/2+ε}) instead of O(T^{1/2}). That still gives the same limit after dividing by τ, so the theorem might be salvageable with a weaker error, but the lemma as stated is not established. The second is the foundational formula (3.1), which is cited to an unpublished preprint (arXiv:2304.09267). The paper is not self-contained at its base. A referee cannot check the main input without chasing an unreviewed source.

The conclusion is that the paper is a minor variation on the author's own prior work, with the advertised connection to FLT being a restatement rather than an insight. The level of rigor is below what I'd expect for a serious contribution, and the main new step relies on an estimate that is neither proved nor referenced. I would not send it to review; it's a desk-reject candidate. If the author can supply a proof of (3.10) or adjust the error term appropriately, it might become a footnote in the Jacob's-ladders program, but as it stands it does not support its claims.

Regards.

Referee Report

4 major / 6 minor

Summary. The paper claims to produce sum-variants of the author's earlier 'ζ-functional' built from an almost linear formula for the Hardy–Littlewood integral. It partitions the interval (T, ¹T(T)) at zeros of Z(t), replaces the integral of Z²(t) by a sum of zero-to-zero areas, then by rectangle areas using the mean-value theorem, and, under the Riemann hypothesis, by rectangles at Bonnet points. Substituting T = xτ/(1−c) yields limit functionals whose value is x; for Fermat rationals this gives the 'ζ-condition' that the limit is not 1, which is called a ζ-equivalent of Fermat–Wiles. The main analytic step is Lemma 1, which asserts that the zero-to-zero sum equals (1−c)T + O(√T), derived from the author's formula (3.1) and an unproved zero-spacing estimate (3.10).

Significance. If the main lemma were proved, the paper would provide a discrete/rectangular variant of the Hardy–Littlewood asymptotic that could be used to state Fermat's Last Theorem as a limit condition. However, the central derivation rests on the estimate (3.10), which is asserted without proof and is stronger than what is known unconditionally; moreover, the final 'ζ-equivalents' are direct logical restatements of Fermat–Wiles once the limit is identified with the Fermat rational. The paper contains no machine-checked proofs or parameter-free derivations beyond the self-cited prior formula (3.1). The algebraic organization is straightforward, but the manuscript as written does not establish its headline claims.

major comments (4)

[Sec. 3.2, Eq. (3.10)] The estimate γ₁−T, ¹T−γ_{N+1}=O(T^{1/6}) is stated without proof or reference. The best known unconditional zero-density/local-density results for ζ(1/2+it) give zeros in intervals of length T^{27/82+ε}, not T^{1/6}; with Z²=O(T^{1/3+ε}) the edge integrals are only O(T^{163/246+ε}), which is o(T) but not O(√T). Consequently Lemma 1, Eq. (3.12), is not established as stated. This is load-bearing because all subsequent theorems (Theorems 1, 2, 5, 6, 7) rely on (3.12). The limit theorems would still follow from a weaker o(T) error, but the manuscript does not provide that weaker version.
[Sec. 3.1, Eq. (3.1)] The fundamental asymptotic ∫_T^{¹T} |ζ(1/2+it)|² dt = (1−c)T + O(T^{1/3+δ}) is cited to [7], an arXiv preprint, with no proof in the present paper. Since every sum-variant and every ζ-condition in §§4–5 reduces to this single input, the paper's results are no stronger than an unpublished source. The author should either prove (3.1) or cite a peer-reviewed version with proof; as it stands, the main conclusion is conditional on an unverified prior result.
[Sec. 4.2, Eq. (4.5)] Theorem 2 is logically a restatement of Fermat–Wiles. By Corollary 1, Eq. (4.4), the limit equals (x^m+y^m)/z^m, so the condition 'limit ≠ 1' is exactly x^m+y^m ≠ z^m. Thus the ζ-condition carries no number-theoretic information beyond the identity supplied by the limit formula. The same applies to Theorems 4 and 7. The paper should explicitly state that the Fermat–Wiles equivalence is a formal restatement and explain what additional analytic insight, if any, the sum-form or pulse-form provides; otherwise the central novelty collapses to a notational reformulation.
[Sec. 5.1, Eqs. (5.1)–(5.3)] The derivation of Property 3 (one maximum of |Z(t)| between successive zeros) invokes formula (5.1) from [2] but does not discuss whether the O(1/t) error is uniform enough to guarantee that Z′/Z is decreasing on every interval (γ′,γ″). This monotonicity is needed for the Bonnet-point construction in Lemma 3 and Theorem 5. A precise statement of the needed uniformity and a complete proof (or an exact reference to a theorem in [2]) should be supplied.

minor comments (6)

[Eq. (3.6)] The symbol γ₀ appears in (3.6) but is not introduced in (3.2)–(3.4); it should be defined explicitly as the largest zero of Z(t) not exceeding T.
[Eq. (3.10)] The phrase 'elementary estimates' is misleading: the zero-gap claim is a deep unproved assertion. If it is intended as a hypothesis, it should be stated as such.
[Remark 4] The cosmological remarks about 'mathematical models of Universe' are outside the mathematical content and should be removed or moved to a separate non-technical note.
[References [7], [8]] References [7] and [8] are arXiv preprints; [8] lacks the arXiv identifier and journal data. The author should provide published versions or more complete citation information.
[Property 6, Eq. (6.2)] The ratio Z²(t₀(n))/Z²(¯t(n)) = (γ_{n+1}−γ_n)/(ξ₂(n)−ξ₁(n)) is derived by equating two expressions for the same integral; the assertion that this ratio is 1 is only true because both sides equal that common area. The presentation overstates the 'conservation law'.
[Sec. 6, Property 7] The quantity K(n,T) in (6.3) is never used; either connect it to a subsequent result or delete it.

Circularity Check

2 steps flagged · score 7.0 of 10

Self-cited formula (3.1) is the load-bearing input; the Fermat-Wiles 'ζ-equivalents' are tautological restatements.

self citation load bearing [Section 3.1, formula (3.1) and Lemma 1 (3.12)]
"Let us remind our almost linear formula, [7], (3.4), (3.6), ∫_{T}^{¹T} |ζ(1/2+it)|² dt = (1−c)T + O(T^{1/3+δ}), T→∞"

Lemma 1, the starting point of every sum-functional and ζ-equivalent in Sections 4–5, is obtained by splitting this integral over zero intervals. The formula itself is not proved in the present paper; it is imported from the author's earlier preprint [7]. Thus the central asymptotic reduces to an unverified self-citation, and the later 'sum-variants' are rearrangements of that same input.
renaming known result [Section 4.2, Corollary 1 (4.4) and Theorem 2 (4.5)]
"Consequently ... lim_{τ→∞} (1/τ) { Σ ... } = (x^m+y^m)/z^m ... Theorem 2. The ζ-condition ... ≠ 1 on the class of all Fermat's rationals expresses the new ζ-equivalent of the Fermat-Wiles theorem."

Theorem 1 proves that the limit equals the fixed x. Setting x = (x^m+y^m)/z^m makes the limit equal to that Fermat rational. Hence the ζ-condition 'limit ≠ 1' is logically identical to 'x^m+y^m ≠ z^m', which is exactly the Fermat-Wiles statement. The 'new ζ-equivalent' is therefore a notational restatement of the known theorem, not an independent zeta-theoretic prediction.

full rationale

The paper is not self-contained: its foundational almost-linear formula (3.1) is a self-citation from [7], and all later results depend on it. The final Fermat-Wiles 'ζ-equivalents' reduce to a tautology because the ζ-functional is constructed so that its limit equals the argument x; for Fermat rationals the inequality 'limit ≠ 1' is literally 'x^m+y^m ≠ z^m'. This is a renaming rather than a derivation. Separately, the load-bearing estimate (3.10), called 'elementary' with no proof, is not a known unconditional O(T^{1/6}) zero-gap bound; known results give T^{1/6+ε}, which would break the O(√T) error in Lemma 1. That is a correctness risk, not a circularity step. Overall, the central claims reduce to a self-citation chain plus a definitional restatement of Fermat-Wiles, giving a partial circularity score of 7.

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

The paper assumes the existence and asymptotic of Jacob's ladders (from the author's prior work), the almost linear increments formula (3.1), and an unproved bound on zero spacing. No free parameters are fitted; the constant c is Euler's constant.

assumptions (5)

domain assumption Existence of the Jacob's ladder φ₁(T) and reverse iterations such that ∫_T^{¹T} Z² dt ~ (1−c)T
Assumed from Moser [4]–[8]; the paper does not derive φ₁.
domain assumption Almost linear increments formula (3.1): ∫_T^{¹T} Z² dt = (1−c)T + O(T^{1/3+δ})
Imported from [7]; central input of the paper, not proved here.
ad hoc to paper Edge zero spacing γ₁ − T, ¹T − γ_{N+1} = O(T^{1/6})
Stated in (3.10) without proof; required to make edge integrals O(√T); not an established result.
domain assumption Riemann hypothesis for the rectangle-pulse representation (Sections 5–6)
Assumed to ensure Z² has one maximum between zeros, used with the Bonnet mean value theorem.
domain assumption Z'/Z decreasing on intervals between zeros under RH (from [2])
Used to justify the uniqueness of t₀(n) in (5.7).

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Cite this review

Pith. "Pith review of Jacob's ladders and new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theory based on sums of elementary $\zeta$-pulses." pith.science (2026). https://pith.science/paper/SNWNMSMY

@misc{pith2026260717731,
  author       = {Pith},
  title        = {Pith review of: Jacob's ladders and new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theory based on sums of elementary $\zeta$-pulses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SNWNMSMY}},
  note         = {Machine review of arXiv:2607.17731}
}

abstract

In this paper we obtain some sum-variants of our first $\zeta$-equivalent of the Fermat-Wiles theorem. For example, on the Riemann hypothesis, we give a new type of $\zeta$-equivalent based on a sum of isolated rectangle-shaped signals (pulses).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 6 linked inside Pith

[7]

Moser, Jacob’s ladders, existence of almost linear increments of the Hardy-Littlewood integral and new types of multiplicative laws, arXiv: 2304.09267

J. Moser, Jacob’s ladders, existence of almost linear increments of the Hardy-Littlewood integral and new types of multiplicative laws, arXiv: 2304.09267

arXiv
[2]

Moser, On some properties of Riemann zeta-function on the critical line, Acta Arith

J. Moser, On some properties of Riemann zeta-function on the critical line, Acta Arith. 25, 33–39, (1974), (in Russian)

1974
[1]

Hardy, J.E

G.H. Hardy, J.E. Littlewood, Contribution to the theory of the Riemann zeta-function and the theory of the distribution of Primes, Acta Math. 41 (1), 119 – 196, (1918)

1918
[3]

Moser, Riemann hypothesis and some infinite set of microscopic universes of the Einstein’s type in the early period of the evolution of the Universe, arXiv: 1307.1095v2

J. Moser, Riemann hypothesis and some infinite set of microscopic universes of the Einstein’s type in the early period of the evolution of the Universe, arXiv: 1307.1095v2

arXiv
[4]

Moser,‘Jacob’s ladders and almost exact asymptotic representation of the Hardy- Littlewood integral‘, Math

J. Moser,‘Jacob’s ladders and almost exact asymptotic representation of the Hardy- Littlewood integral‘, Math. Notes 88, (2010), 414-422, arXiv: 0901.3937

arXiv 2010
[5]

Moser, ‘Jacob’s ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc

J. Moser, ‘Jacob’s ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc. Steklov Inst. 276 (2011), 208-221, arXiv: 1103.0359

arXiv 2011
[6]

Moser, Jacob’s ladders, reverse iterations and new infinite set ofL 2-orthogonal systems generated by the Riemannζ 1 2 +it -function, arXiv: 1402.2098v1

J. Moser, Jacob’s ladders, reverse iterations and new infinite set ofL 2-orthogonal systems generated by the Riemannζ 1 2 +it -function, arXiv: 1402.2098v1

arXiv
[8]

J. Moser, Jacob’s ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg formula (1946) and the Fermat-Wiles theorem, arXiv: 2312.12085

arXiv 1918

Show all 9 references

[9]

Kano (edit.),Georg Friedrich Bernhard Riemann, 1991, ISBN4-535-78181-8 C3041 P3800E, (in Japanese, our results are situated on pp

K. Kano (edit.),Georg Friedrich Bernhard Riemann, 1991, ISBN4-535-78181-8 C3041 P3800E, (in Japanese, our results are situated on pp. 127-132). Department of Mathematical Analysis and Numerical Mathematics, Comenius Uni- versity, Mlynska Dolina M105, 842 48 Bratislava, SLOV AK...

1991

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [7]

Moser, Jacob’s ladders, existence of almost linear increments of the Hardy-Littlewood integral and new types of multiplicative laws, arXiv: 2304.09267

J. Moser, Jacob’s ladders, existence of almost linear increments of the Hardy-Littlewood integral and new types of multiplicative laws, arXiv: 2304.09267

arXiv

[2] [2]

Moser, On some properties of Riemann zeta-function on the critical line, Acta Arith

J. Moser, On some properties of Riemann zeta-function on the critical line, Acta Arith. 25, 33–39, (1974), (in Russian)

1974

[3] [1]

Hardy, J.E

G.H. Hardy, J.E. Littlewood, Contribution to the theory of the Riemann zeta-function and the theory of the distribution of Primes, Acta Math. 41 (1), 119 – 196, (1918)

1918

[4] [3]

Moser, Riemann hypothesis and some infinite set of microscopic universes of the Einstein’s type in the early period of the evolution of the Universe, arXiv: 1307.1095v2

J. Moser, Riemann hypothesis and some infinite set of microscopic universes of the Einstein’s type in the early period of the evolution of the Universe, arXiv: 1307.1095v2

arXiv

[5] [4]

Moser,‘Jacob’s ladders and almost exact asymptotic representation of the Hardy- Littlewood integral‘, Math

J. Moser,‘Jacob’s ladders and almost exact asymptotic representation of the Hardy- Littlewood integral‘, Math. Notes 88, (2010), 414-422, arXiv: 0901.3937

arXiv 2010

[6] [5]

Moser, ‘Jacob’s ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc

J. Moser, ‘Jacob’s ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc. Steklov Inst. 276 (2011), 208-221, arXiv: 1103.0359

arXiv 2011

[7] [6]

Moser, Jacob’s ladders, reverse iterations and new infinite set ofL 2-orthogonal systems generated by the Riemannζ 1 2 +it -function, arXiv: 1402.2098v1

J. Moser, Jacob’s ladders, reverse iterations and new infinite set ofL 2-orthogonal systems generated by the Riemannζ 1 2 +it -function, arXiv: 1402.2098v1

arXiv

[8] [8]

J. Moser, Jacob’s ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg formula (1946) and the Fermat-Wiles theorem, arXiv: 2312.12085

arXiv 1918