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Jacob's ladders, $\zeta$-transformation of the Fourier orthogonal system (2014) and new infinite sets of $\zeta$-equivalents of the Fermat-Wiles theorem

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Pith's one-line read The paper claims that Fermat's Last Theorem is equivalent to infinitely many conditions that certain zeta-weighted limits are not equal to 1.

desk verdict The new ζ-equivalents are definitional restatements: Theorem 1 forces the limit to equal x^k, so substituting the Fermat rational makes the condition exactly x^n+y^n≠z^n. read the letter →

arxiv 2507.06724 v1 pith:AHG6O4BS submitted 2025-07-09 math.NT

Jan Moser This is my paper
classification math.NT MSC 11M0611D41
keywords Jacob'sladdersRiemannzetafunctionFermat-WilestheoremHardy-LittlewoodintegralorthogonalsystemscriticallineFouriersystemzeta-equivalents
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

This paper claims that Fermat's Last Theorem can be reformulated as an infinite family of statements about Riemann's zeta function on the critical line. For each positive integer $k$ and each member of any continuous $L^2$-orthogonal system, the author defines a zeta-weighted integral over a 'ladder-transformed' interval and proves that its normalized limit equals $x^k$. Substituting $x=(x^n+y^n)/z^n$, the limit equals exactly 1 if Fermat's equation has a solution; since Fermat-Wiles says no such solution exists for $n\ge 3$, the condition 'limit not equal to 1' becomes a new $\zeta$-equivalent of the theorem. The proof rests on quoted results about Jacob's ladders and the $\zeta$-transformation of orthogonal systems, which are not proved in this paper.

What carries the argument

The load-bearing object is Jacob's ladder $\varphi_1(t)$, a function associated with the integral of $|\zeta(\tfrac12 + it)|^2$ whose reverse iterations $\varphi_1^{-r}(T)=r/T$ produce a nearly equidistant partition of $[T, {}^k T]$. The second piece is the $\zeta$-transformation theorem quoted from [4], which converts any continuous $L^2$-orthogonal system $\{f_m\}$ into weighted orthogonal systems carrying the product $\prod_{r=0}^{k-1}|\zeta(\tfrac12 + i\varphi_1^r(t))|^2$. The argument's engine is the normalization identity (1.6) and Lemma 2, which assert that the weighted integral equals $(1+o(1)) A_m \ln^k T$; the substitution $T = \exp(x\,\ln\tau/(k\sqrt{A_m}))$ converts that asymptotic into the sharp limit $x^k$.

What would settle it

Compute the left side of (4.2) for a single concrete instance, say $k=1$, $m=1$, $f_1(t)=1$, $l=\tfrac12$, $x=2$, using a numerical realization of $\varphi_1$ derived from the Hardy–Littlewood integral; if the normalized integral does not approach $2$ as $\tau\to\infty$, Theorem 1 fails. A purely arithmetic falsifier would be any triple $x,y,z,n\ge3$ with $x^n+y^n=z^n$, which would force the limit (4.3) to equal $1$ and contradict Theorem 2.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2: for every Fermat rational $x = (x^n + y^n)/z^n$ with $n \ge 3$, the normalized zeta-weighted limit in (4.3) is not equal to $1$, and this '$\ne 1$' condition is a new $\zeta$-equivalent of the Fermat-Wiles theorem. The reason is Theorem 1: the same kind of limit equals $x^k$ exactly for every fixed $k,m$ and any continuous $L^2$-orthogonal system, after the substitution $T = \exp(x\,\ln\tau/(k\sqrt{A_m}))$. Because a Fermat solution would make $x=1$, the inequality is exactly the negation of Fermat's equation. The paper then specializes to the elementary Fourier orthogonal system, obtaining explicit infinite families (Theorems 4 and 6) indexed by $k$ and $m$.

Load-bearing premise

The load-bearing premise is that a special function called Jacob's ladder exists with the precise growth and partition properties stated in (2.1)–(2.6), and that the quoted theorem transforming orthogonal systems is valid; if either is false or unproved, the claimed limit identities and the equivalence to Fermat's theorem do not follow.

Editorial extensions

If this is right

  • If the central identities hold, Fermat's Last Theorem is equivalent to any of infinitely many statements of the form 'the limit in (4.3) is not 1,' with independent parameters $k$, $m$, and the choice of orthogonal system.
  • Theorem 1 supplies a family of exact integral representations of $x^k$ in terms of zeta values on the critical line, for all $x>0$.
  • For the Fourier system, Theorems 4 and 6 give explicit infinite sets of $\zeta$-equivalents, including the dual pair (1.17) and (1.19).
  • The difference identities (5.7) and (5.8) show that orthogonality relations among the transformed trigonometric functions survive in the limit, so the usual Fourier normalization constants reappear as zeta-weighted limits.

Reading between the lines

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

  • The author leaves implicit that the same substitution $x \mapsto r$, for any rational $r \neq 1$, would turn other 'no integer solutions' statements into $\zeta$-conditions, provided the ladder machinery is valid.
  • A natural testable extension is to verify Lemma 2 numerically for small $k$ and fixed $m,l$; success would lend independent support to the ladder asymptotics, while failure would locate the breakdown in (2.1)–(2.6) rather than in the Fermat step.
  • The dual construction in Remark 4 suggests that each pair of orthogonal functions with distinct normalization constants generates paired equivalents, so the family of $\zeta$-equivalents is at least as rich as the orthogonal system chosen.
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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

3 major / 5 minor

Summary. The manuscript continues the author's series on Jacob's ladders and claims to produce new infinite sets of ζ-equivalents of the Fermat-Wiles theorem. It recalls an orthogonality transformation for L2-orthogonal systems from [4], derives Lemma 2 (Eq. 3.18) as an asymptotic identity for a weighted integral of |ζ(1/2+it)|², and then substitutes T = exp(x / A_m^{1/k} ln τ) to obtain Theorem 1 (Eq. 4.2), a limit equal to x^k. Substituting x = (x^n+y^n)/z^n gives Theorem 2 (Eq. 4.3), which is presented as a new ζ-equivalent of Fermat's Last Theorem. Section 5 treats the Fourier cosine/sine system analogously, with Theorems 3-6 obtained by formally replacing l by the Fermat rational. The derivation from Lemma 2 to Theorem 1 is a straightforward substitution, but the central claim that these are new equivalents is not supported.

Significance. If the quoted Jacob's-ladder machinery from [4] and [9] were fully established, Lemma 2 and Theorem 1 would be a correct conditional asymptotic identity. The paper is transparent that the core inputs are quoted, and the formulas are explicit rather than fitted. However, the claimed ζ-equivalents of the Fermat-Wiles theorem are obtained by a substitution that makes the stated condition a restatement of the inequality q ≠ 1; no property of ζ beyond the asymptotic identity, and no arithmetic information, is used. The infinite families indexed by k collapse because q^k ≠ 1 is equivalent to q ≠ 1 for q > 0. Thus, even if all quoted analysis is accepted, the paper does not establish a new connection between the zeta function and Fermat's Last Theorem.

major comments (3)
  1. [Sec. 4, Thm. 1 and Thm. 2] Theorem 2 is a tautological restatement of Fermat's Last Theorem. By Theorem 1, Eq. (4.2), the normalized integral tends to x^k for every x > 0. Setting x = q = (x^n + y^n)/z^n, the 'ζ-condition' in (4.3) is q^k ≠ 1. Since q > 0 and k ∈ N, q^k = 1 iff q = 1, so the condition is equivalent to (x^n + y^n)/z^n ≠ 1, i.e., to x^n + y^n ≠ z^n. No information about ζ enters except through (4.2), which holds for arbitrary x, k, m and f_m. The same construction with any positive rational q in place of the Fermat rational would produce an equally valid 'ζ-equivalent' of the corresponding Diophantine inequality. The claimed infinite family also collapses, because q^k ≠ 1 is independent of k.
  2. [Sec. 5, Thms. 3-6] The same substitution mechanism appears in Section 5. Theorem 3 (5.1) gives the limit 2l; replacing l by q = (x^n + y^n)/z^n turns the condition (5.3) into 2q ≠ 2, i.e., q ≠ 1. Likewise, Theorem 5 (5.4) gives the limit l, so the condition (5.6) becomes q ≠ 1. These are not independent ζ-equivalents; they are the same Fermat inequality written with different normalizing constants. The Fourier system and the zeta function enter only through the normalization identities (1.13)-(1.15), which are used to set the constants, and no new arithmetic content is extracted.
  3. [Sec. 2 and Eq. (3.18)] The central asymptotic (3.18) rests on (3.2), which is the mean-value form of the quoted orthogonality transformation (1.2)-(1.3) from [4], and on the estimate (3.16) for the product of ω[φ_1^r(α)]. Both depend on the existence of Jacob's ladder φ_1 and on the geometrical and asymptotic properties (2.1)-(2.6), including the almost-linear increment formula (2.6) quoted from [9]. None of these inputs is proved in this manuscript. Since every theorem in Sections 4 and 5 depends on Lemma 2, the paper's conclusions are conditional on a substantial external framework. This dependence should be stated explicitly at the point of use, with theorem-level references, before the results can be assessed as self-contained.
minor comments (5)
  1. [Eq. (6.2)] The displayed formula in (6.2) is malformed: the integrand contains a bare 'd' where 'dt' is intended, and the parentheses/brackets are unbalanced. Please correct the typesetting.
  2. [References [7] and [8]] References [7] and [8] are both given the arXiv identifier 2312.12085; if these are different papers, one of the identifiers must be wrong, and if they are the same paper, the duplication should be removed.
  3. [Eqs. (1.9), (4.2), (5.1)] The notation for the integration limits using expressions such as [W(x,τ)+2l]^k is not introduced; the relationship between W, the Jacob's-ladder endpoints, and the exponent k should be spelled out with explicit lower and upper limits.
  4. [Notation in (1.10)] The symbol k√A_m is used without definition; if it denotes the k-th root of A_m, this should be stated explicitly, since the subsequent cancellation in the limit depends on that interpretation.
  5. [Throughout] There are several typographical errors, including 'followig' before (1.13), 'Fuctionals' in the heading of Section 5, and 'differents' in Remark 8; these should be corrected.

Circularity Check

2 steps flagged · score 10.0 of 10

The new 'ζ-equivalents' are tautological: Theorem 1 forces the limit to equal the chosen Fermat rational power, so Theorem 2's condition 'limit ≠ 1' is just the Fermat inequality restated.

  1. self definitional [Section 4.1, Eqs. (4.1)-(4.3), with W(x,τ) defined in (1.10)]
    "W (x, τ) = W (x, τ; k, m) = exp{ x/(k√Am) ln τ } ... lim_{τ→∞} 1/ln^k τ ∫_{[W(x,τ)+2l]^k}^{[W(x,τ)]^k} f_m^2(φ_1^k(t)-W(x,τ))× ∏_{r=0}^{k-1}|ζ(1/2+iφ_1^r(t))|^2 dt = x^k ... Next, the substitution x → (x^n+y^n)/z^n into (4.2) gives ... ζ-condition ... ≠ 1"

    Equation (4.2) is obtained from Lemma 2 (3.18), which states that the unnormalized integral is A_m ln^k T asymptotically. The choice T = W(x,τ) with ln W = x ln τ/(k√A_m) is what makes the τ-normalized limit come out to x^k; the value x^k is placed into the construction via the definition of W, not extracted from any property of ζ. Substituting x = (x^n+y^n)/z^n, the 'ζ-condition' in (4.3) is precisely 'limit ≠ 1', which by (4.2) means (x^n+y^n)/z^n ≠ 1, i.e. x^n+y^n ≠ z^n. That is exactly the Fermat statement being 'equivalently' expressed. Any positive rational q would produce an equally valid 'ζ-equivalent' by using W(q,τ), so the claim carries no zeta-function information.

  2. self definitional [Section 5.1, Theorems 4 and 6, Eqs. (5.3) and (5.6)]
    "in the case l → (x^n+y^n)/z^n ... The ζ-condition lim_{T→∞} 1/ln^k T ∫_{[T+2 (x^n+y^n)/z^n]^k}^{^k T} ∏_{r=0}^{k-1}|ζ(1/2+iφ_1^r(t))|^2 dt ≠ 2 ... expresses the next ζ-equivalent of the Fermat-Wiles theorem"

    Theorem 3 (5.1) establishes that the same normalized integral tends to 2l. Replacing l by the Fermat rational q = (x^n+y^n)/z^n gives the limit 2q, so the reported condition '≠ 2' is logically equivalent to q ≠ 1, i.e. to x^n+y^n ≠ z^n. The choice of integration endpoint / length parameter is exactly what fixes the limit value, so the stated ζ-equivalent is again the Fermat inequality rephrased in terms of a constructed limit, not a new connection involving ζ.

full rationale

The paper's central new claim (Theorem 2) reduces to FLT by construction. Theorem 1's limit identity (4.2) is manufactured by defining W(x,τ) with ln W proportional to x ln τ, so the normalized zeta-integral is forced to converge to x^k. Substituting x = (x^n+y^n)/z^n makes the ζ-condition 'limit ≠ 1' exactly equivalent to (x^n+y^n)/z^n ≠ 1, i.e. to the Fermat inequality x^n+y^n ≠ z^n. The same pattern is repeated in Theorems 4 and 6, where l is replaced by the Fermat rational and the limit value is thereby set to a multiple of q. Thus the 'new infinite sets of ζ-equivalents' are not derived from zeta-function information; they are the Fermat statement renamed through an engineered limit. The infrastructure also relies on unproved Jacob's-ladder properties and an orthogonality transformation cited from the author's earlier papers [4] and [9], so the non-tautological part of the argument is not self-contained, but even granting that infrastructure the main equivalence is definitional. The paper adds no independent evidence about ζ or about FLT; score 10.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central computation relies on a chain of self-cited results (the ladder's existence, the orthogonality theorem, and the almost-linear increments). No parameters are fitted to data; instead, the free variable x is set to the Fermat rational, turning a parametric identity into an equivalence statement.

assumptions (5)
  • domain assumption Existence of Jacob's ladder φ_1(t) with the asymptotic and geometric properties in Section 2.
    Quoted from author's earlier papers, e.g., [2], [5], [9]; never proved in this text.
  • domain assumption Orthogonality transformation theorem (1.2)-(1.3) for arbitrary L²-orthogonal systems.
    Quoted from [4]; the foundation for (1.6), (3.18) and all theorems.
  • domain assumption Almost linear increments of the Hardy-Littlewood integral (2.6).
    Cited from [9]; used to justify reverse-iteration geometry and Lemma 1.
  • domain assumption The segment parameter l is fixed and independent of T, satisfying l = o(T/ln T).
    Stated in Remark 2; needed for the mean-value estimates (3.10)-(3.16).
  • standard math Standard mean value theorem and asymptotic expansion manipulations.
    Used in Section 3 to estimate ω[φ_r^1(α)] and derive Lemma 1.
invented entities (3)
  • Jacob's ladder function φ_1(t)
    purpose: Transforms the Hardy-Littlewood integral into almost linear increments and generates new orthogonal systems.
    Introduced and developed in the author's series; no independent evidence or falsifiable prediction outside the theory.
  • Reverse iterations of Jacob's ladders (φ_1^{-r}(T) = r-th iterated point)
    purpose: Define the integration intervals [k/T, k/T+2l] used in all limits.
    Derived from φ_1; no independent confirmation.
  • ζ-transformed Fourier orthogonal system (1.12)
    purpose: Provides the functions whose normalization integrals produce the new ζ-equivalents.
    A construction built from the zeta function and Jacob's ladders; no external evidence.

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Cite this review

Pith. "Pith review of Jacob's ladders, $\zeta$-transformation of the Fourier orthogonal system (2014) and new infinite sets of $\zeta$-equivalents of the Fermat-Wiles theorem." pith.science (2026). https://pith.science/paper/AHG6O4BS

@misc{pith2026250706724,
  author       = {Pith},
  title        = {Pith review of: Jacob's ladders, $\zeta$-transformation of the Fourier orthogonal system (2014) and new infinite sets of $\zeta$-equivalents of the Fermat-Wiles theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHG6O4BS}},
  note         = {Machine review of arXiv:2507.06724}
}
abstract

In this paper we obtain new infinite sets of $\zeta$-equivalents of the Fermat-Wiles theorem based on the elementary Fourier orthogonal system, Riemann's zeta-function and Jacob's ladders.

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Works this paper leans on

17 extracted references · 17 canonical work pages

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