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Jacob's ladders, E. C. Titchmarsh's hypothesis (1934) and new $\zeta$-equivalents of the Fermat-Wiles theorem or connections between Fermat's rationals and the Gram's sequence

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Pith's one-line read The paper claims that three explicit zeta-limit conditions are new equivalents of the Fermat-Wiles theorem.

desk verdict The zeta-conditions are constructed to equal (x^n+y^n)/z^n exactly, so the 'new equivalents' are the Fermat inequality restated in zeta-integral language. read the letter →

arxiv 2506.01706 v1 pith:OQVKIM2F submitted 2025-06-02 math.NT

Jan Moser This is my paper
classification math.NT MSC 11M0611D41
keywords Riemannzeta-functionFermat-WilestheoremGrampointsTitchmarshsumJacob'sladdersHardy-LittlewoodintegralSelbergformulazeta-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 certain limit expressions built from the Riemann zeta-function are exact restatements of Fermat's Last Theorem. Each expression is a product of zeta integrals divided by a Gram-point sum, and the paper's asymptotic formula forces the limit to equal a prescribed positive number $x$. When $x$ is specialised to a Fermat rational $(x^n+y^n)/z^n$, the statement that the limit is not $1$ is then equivalent to Fermat's Last Theorem. The construction links zeta-function theory through Gram points and Jacob's ladders to the arithmetic of integer powers.

What carries the argument

The load-bearing machinery is the 1991 asymptotic formula (3.2) for Titchmarsh's sum over the Gram sequence $(t_\nu)$, the points where $\vartheta(t_\nu)=\pi\nu$: $$\sum_{T\le t_\nu\le 2T} |\zeta(\tfrac12+it_\nu)|^2|\zeta(\tfrac12+it_{\nu+1})|^2 = \frac{3}{4\$pi^{5}$}T\$ln^{5}$ T\,(1+O(1/\ln T)).$$ This is combined with quotient formulas, such as (3.1), that express $\ln T$ through integrals of $|\zeta(\tfrac12+it)|^2$ and $|\zeta(\sigma+it)|^2$ over reverse-iteration intervals of what the paper calls Jacob's ladders, a family of reverse iterations $\varphi_1^{-r}(T)=r/T$ that partition the Hardy-Littlewood integral into asymptotically equal parts. The substitution $T=\frac{4\pi^5}{3\zeta^5(2\sigma)}x\tau$ makes the cross-bred product of five integrals and the Gram sum equal $x$, converting the asymptotic formula into an exact limit identity.

What would settle it

An independent proof or numerical computation of $\sum_{T\le t_\nu\le 2T} |\zeta(\tfrac12+it_\nu)|^2|\zeta(\tfrac12+it_{\nu+1})|^2$ for large $T$ that disagrees with $(3/(4\pi^5))T\ln^5 T$ would invalidate the chain; likewise, a direct evaluation of the limit in (3.6) for any fixed $x>0$ that returns a value different from $x$ would refute the claimed identity.

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Extended reading notes

Core claim

The central discovery is Theorem 1: for every fixed $\sigma\ge 1/2+\epsilon$, the zeta-condition (3.8) -- that the displayed limit built from $|\zeta(\sigma+it)|^2$, $|\zeta(\tfrac12+it)|^2$, and the Titchmarsh sum over Gram points is not equal to $1$ -- is proposed as a new equivalent of the Fermat-Wiles theorem. The reason is Lemma 2, identity (3.7): the same limit is forced, by the asymptotic formula (3.2) and the substitution $T=\frac{4\pi^5}{3\zeta^5(2\sigma)}\frac{x^n+y^n}{z^n}\tau$, to equal exactly $(x^n+y^n)/z^n$. Hence the inequality in (3.8) says that $x^n+y^n\neq z^n$ for positive integers $x,y,z$ and $n\ge 3$. Theorems 3 and 4 repeat the construction with Selberg's $S_1(t)$ integrals and with the fourth-power Gram sum, yielding two further equivalents.

Load-bearing premise

The whole chain rests on the 1991 asymptotic formula for the Titchmarsh sum over Gram points, which the paper cites but does not prove; if its leading term were not $(3/(4\pi^5))T\ln^5 T$, the constructed limits would not equal $x$ and the equivalences would collapse.

Editorial extensions

If this is right

  • If Theorem 1 is correct, Fermat's Last Theorem becomes the assertion that the explicit limit in (3.8) is never equal to 1 for any Fermat rational.
  • Theorem 3 yields a second equivalent in which the integral of $|S_1(t)|^{2l}$ replaces the $\sigma$-integral, for every fixed $l\in\mathbb N$.
  • Theorem 4 yields a third equivalent based on the fourth-power Gram sum $\sum |\zeta(\tfrac12+it_\nu)|^4$ rather than on products of neighbouring Gram values.
  • The chain (6.1) joins the three new functionals to the author's earlier equivalents, giving a continuum of chains as $x$ ranges over an interval.

Reading between the lines

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

  • Editorial extension: since the limit is forced by construction to equal the Fermat rational itself, these equivalences are exact rewrites of Fermat's Last Theorem rather than new analytic estimates; they do not by themselves offer a route to deciding the inequality.
  • Editorial extension: the same device can be applied to any Diophantine condition by substituting the relevant rational expression into $x$; the Fermat case is one instance of a general template.
  • Editorial extension: the weak point of the whole chain is the unproved asymptotic formula (3.2), so independently testing or proving that formula is the natural first check of the claimed equivalences.
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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 / 6 minor

Summary. This paper (arXiv:2506.01706) claims to produce three new 'zeta-equivalents' of the Fermat-Wiles theorem. The author starts from an asymptotic formula for the Titchmarsh sum over Gram points, combined with a quotient formula for integrals of |zeta(1/2+it)|^2 and |zeta(sigma+it)|^2, to obtain a functional identity F(x)=x (Lemma 1). Substituting x=(x^n+y^n)/z^n yields Lemma 2, from which Theorem 1 asserts that a certain limit expression being different from 1 is a zeta-equivalent of FLT. Sections 4 and 5 produce analogous statements using Selberg's formula and a fourth-moment Gram sum. Section 6 continues a chain of formal equivalences from earlier preprints.

Significance. At most the paper shows that, assuming the quoted asymptotics, a particular zeta-integral/Gram-sum limit is algebraically equal to a prescribed positive number x. Since the result is constructed by substitution, the claimed equivalence to FLT is tautological: the condition 'limit != 1' is exactly '(x^n+y^n)/z^n != 1'. Thus the paper does not present a substantive new reformulation of FLT. The computations are explicit and the logic from the quoted formulas to the final identities is transparent, but the central intellectual claim that these are new zeta-equivalents is not supported. The paper also relies on its own cited asymptotic formulas without proof or independent verification.

major comments (3)
  1. [Section 3.2, Theorem 1; Eq. (3.7)] Lemma 2 (Eq. (3.7)) proves that the limiting expression in Theorem 1 equals (x^n+y^n)/z^n identically. Consequently, the condition (3.8) that this limit is not equal to 1 is, by the same equality, exactly the inequality x^n+y^n != z^n. No property of the zeta function beyond the algebraic manipulation of the quoted asymptotic formula is used. The equivalence is therefore a tautological restatement of the Fermat inequality in the language of zeta-integrals and Gram sums. The same criticism applies verbatim to Theorem 3 (Eq. (4.10)) and Theorem 4 (Eq. (5.6)).
  2. [Section 3.1, Eq. (3.2) / Eq. (1.8)] The functional identity F(x)=x, and hence all three theorems, depends on the exact leading term 3/(4 pi^5) T ln^5 T in the Titchmarsh sum (3.2). This formula is cited from the author's own 1991 paper [4] and its proof is not reproduced or independently verified in the manuscript; the 1980 proof of Titchmarsh's hypothesis [3] is also cited without proof. Since the substitution (3.5) is calibrated to this precise constant and power of logarithm, the claimed equivalence would fail if the true asymptotic had a different constant or exponent. A load-bearing input of this kind cannot be left as a reference to the author's previous work.
  3. [Section 2.1, Eqs. (2.1)-(2.2)] The definition of reverse iterations of Jacob's ladders is internally inconsistent in the displayed form: if phi_1^{-r}(T)=r/T as written, then the assertion T ~ ^1T in (2.5) is impossible for T tending to infinity, and phi_1(r/T)=(r-1)/T cannot hold for r>1. This makes the notation [A]_1 used in all subsequent integrals ambiguous, and the reader cannot verify the exact domain of integration in Lemmas 1-4 and Theorems 1, 3, and 4.
minor comments (6)
  1. [Section 3.2] 'wee obtain' is a typo for 'we obtain'.
  2. [Eqs. (3.6), (3.8), (4.9), (4.10), (5.5)] The notation [A]_1 appearing in the integral limits is not defined in the paper; the author should clarify whether it denotes the reverse iteration ^1T and define it consistently with Section 2.
  3. [Eqs. (1.10), (1.12), (3.6), (3.8)] The integral limits are typeset in a way that is very hard to parse; the common substitution factor should be defined once and displayed separately to improve readability.
  4. [References] The bibliography entry [16] spells the author's name 'Titschmarsh' in the reference list; it should be 'Titchmarsh'.
  5. [Remarks 1, 5, and Section 6.2] The philosophical digressions about 'Pythagorean philosophy' and the 'Friedmann-Hubble expanding Universe', as well as the Hardy quotation, do not contribute to the mathematical content and are out of place in a research paper.
  6. [References [5]-[13]] Several of the cited preprints are the author's own works that are not readily available in standard databases; the manuscript should state which specific statements from these preprints are being used.

Circularity Check

3 steps flagged · score 9.0 of 10

Theorem 1's ζ-condition is the Fermat inequality by construction: Lemma 2 fixes the limit to equal (x^n+y^n)/z^n, so '≠1' is just x^n+y^n≠z^n.

  1. self definitional [Section 3.2, Lemma 2 (eq. 3.7) and Theorem 1 (eq. 3.8)]
    "Lemma 2. ... = xn + yn / zn ... Consequently, we have the following result Theorem 1. The ζ-condition ... != 1 ... represents the next ζ-equivalent of the Fermat-Wiles theorem."

    Lemma 2 is obtained by substituting T = (4π^5/(3ζ^5(2σ)))·((x^n+y^n)/z^n)τ into eq. (3.4), so the leading T ln^5 T term cancels and the displayed limit equals (x^n+y^n)/z^n identically. Therefore the 'ζ-condition' in Theorem 1, namely that the same limit is not equal to 1, is exactly (x^n+y^n)/z^n ≠ 1, i.e., x^n+y^n ≠ z^n. The claimed equivalence with Fermat's Last Theorem is built in by the choice of x, not derived from the zeta-function structure; the analytic expression is forced to equal the Fermat rational by construction.

  2. self citation load bearing [Section 1.2(C) and Section 3.1(B), eqs. (1.8) and (3.2)]
    "Next, in 1991, we have derived the following asymptotic formula (1.8) ... = 3/4π^5 T ln^5 T + O(T ln^4 T) (see [4], (2.4), (2.6) and (2.10))."

    Every lemma and theorem in Sections 3–5 is obtained by algebraic rearrangement of this asymptotic formula after substituting T = C·xτ, with the Fermat rational then inserted for x. The formula is quoted from the author's own 1991 paper [4] and is not proved or independently verified in the present paper. Thus the genuinely analytic content of all three 'new zeta-equivalents' resides entirely in a load-bearing self-citation, while the connection to Fermat's Last Theorem is produced by the substitution rather than by any new zeta-function argument.

1 more flagged steps
  1. renaming known result [Sections 4.2–5.1, Theorems 3 and 4 (eqs. (4.10) and (5.6))]
    "Finally, we obtain from (4.9), exactly as we did it in (3.6) – (3.8), the following result. Theorem 3. The ζ-condition ... != 1 ... represents new ζ-equivalent of the Fermat-Wiles theorem."

    Theorems 3 and 4 are produced by literally repeating the same substitution with different constants and integrands: the limit is again made to equal (x^n+y^n)/z^n, so the '≠ 1' conditions are just the Fermat inequality x^n+y^n ≠ z^n rewritten with S1-integrals or fourth powers of |ζ(1/2+itν)|. These are not independent equivalents; they are the same tautological restatement relabeled through the self-cited asymptotic formulae and Selberg's formula.

full rationale

The paper's central derivation is not a substantive new reformulation of Fermat's Last Theorem, because the claimed ζ-condition is manufactured to equal the Fermat rational. Lemma 1 shows that the chosen cross-bred expression is identically x after the substitution (3.5); Lemma 2 then specializes x to (x^n+y^n)/z^n, making Theorem 1's condition 'limit ≠ 1' literally equivalent to x^n+y^n ≠ z^n. The same construction is repeated in Theorems 3 and 4. If the asymptotic formula (3.2)/(1.8), quoted from the author's 1991 paper [4], is accepted, the identities are correct as algebraic consequences, but the 'new zeta-equivalents' carry no additional number-theoretic content beyond substituting the Fermat rational into that self-cited formula. Hence the central claim reduces by definition to its own input: the statement to be proved is embedded in the value chosen for x. Score 9 rather than 10 only because the underlying asymptotic formula, if independently established, would be a genuine analytic result; but the Fermat equivalence itself is tautological.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

No free parameters are fitted in this paper; constants such as 4*pi^5/(3*zeta^5(2*sigma)) are chosen by substitution to cancel the constants in prior asymptotic formulas. The central claim rests on unverified assumptions from the author's earlier works, especially the Titchmarsh-sum asymptotic (3.2), and on Jacob's ladders as a nonstandard, self-developed entity.

assumptions (6)
  • standard math Euler product and analytic continuation of the Riemann zeta function
    Used in (1.1)-(1.3) to define zeta and the Z-signal.
  • standard math Hardy-Littlewood mean-value formulas for |zeta(1/2+it)|^2 and |zeta(sigma+it)|^2
    Invoked in Section 3.1 through [11] as the basis of quotient formula (3.1).
  • domain assumption Titchmarsh's hypothesis (1934) is true with A=4
    Assumed from the author's 1980 paper [3]; no proof is reproduced and no independent verification is offered.
  • domain assumption Asymptotic formula (3.2) for the Titchmarsh sum over Gram points
    Assumed from the author's 1991 paper [4]; it is the load-bearing input for F(x)=x and all subsequent equivalences.
  • domain assumption Existence and properties of Jacob's ladders and almost linear increments (2.6), (4.4)
    Assumed from the author's papers [5]-[9]; used to establish quotient formulas (3.1) and (4.5).
  • domain assumption Selberg's formula (4.1) for integrals of |S1(t)|^(2l)
    Standard result from Selberg [15]; used to build Theorem 3's second equivalent.
invented entities (1)
  • Jacob's ladder function phi_1(t)
    purpose: Defines reverse iterations r/T and intervals used to partition the Hardy-Littlewood integral; supplies almost linear increments on which formulas (3.1) and (4.5) rely.
    Introduced and developed in the author's own prior papers [5]-[9]; no independent verification is provided here or in the cited literature. It is an internal construction whose existence is assumed from those papers.

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Pith. "Pith review of Jacob's ladders, E. C. Titchmarsh's hypothesis (1934) and new $\zeta$-equivalents of the Fermat-Wiles theorem or connections between Fermat's rationals and the Gram's sequence." pith.science (2026). https://pith.science/paper/OQVKIM2F

@misc{pith2026250601706,
  author       = {Pith},
  title        = {Pith review of: Jacob's ladders, E. C. Titchmarsh's hypothesis (1934) and new $\zeta$-equivalents of the Fermat-Wiles theorem or connections between Fermat's rationals and the Gram's sequence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQVKIM2F}},
  note         = {Machine review of arXiv:2506.01706}
}
abstract

In connection of our proof (1980) of the Titchmarsh's hypothesis (1934), we have obtained two asymptotic formulae (1991). In this paper we obtain three new $\zeta$-equivalents of the Fermat-Wiles theorem based on the mentioned asymptotic formulae.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.NT 2025-07 reject novelty 4.0 of 10

    A zeta-weighted integral limit is shown to equal (x^n+y^n)/z^n exactly, making 'limit not equal 1' a verbatim restatement of Fermat's Last Theorem.

Reference graph

Works this paper leans on

16 extracted references · 7 canonical work pages · cited by 1 Pith paper

  1. [4]

    On the order of the Titchmarsh's sum in the theory of the Riemann zeta-function and on the biquadratic effect in the information theory

    J. Moser, On the order of Titchmarsh’s sum in the theory of the Riemann zeta-function, Czechoslovak Math. J., 41 (116), (1991), 663 – 684, (in Russian), arXiv: 1112.5948v1

    work page Pith review arXiv 1991

  2. [3]

    Moser, Proof of the Titchmarsh’s hypothesis in the theory of the Riemann zeta-function, Acta Arit., 36, (1980), 147 – 156, (in Russian)

    J. Moser, Proof of the Titchmarsh’s hypothesis in the theory of the Riemann zeta-function, Acta Arit., 36, (1980), 147 – 156, (in Russian)

    work page 1980

  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. [2]

    G. H. Hardy, A Mathematician ’s Apology, Cambridge Univ. Press, (1940)

    work page 1940

  5. [5]

    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. [6]

    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. [7]

    Moser, Jacob’s ladders, reverse iterations and new infinite set of L2-orthogonal systems generated by the Riemann ζ 1 2 + it -function, arXiv: 1402.2098v1

    J. Moser, Jacob’s ladders, reverse iterations and new infinite set of L2-orthogonal systems generated by the Riemann ζ 1 2 + it -function, arXiv: 1402.2098v1

    arXiv

  8. [8]

    Moser, Jacob’s ladders and vector operator producing new generations of L2-orthogonal systems connected with the Riemann’s ζ 1 2 + it -function, arXiv: 2302.0750.v3

    J. Moser, Jacob’s ladders and vector operator producing new generations of L2-orthogonal systems connected with the Riemann’s ζ 1 2 + it -function, arXiv: 2302.0750.v3

    arXiv

Show all 16 references
  1. [9]

    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. [10]

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

    1918 arXiv

  3. [11]

    J. Moser, Jacob’s ladders and new equivalents of the Fermat-Wiles theorem connected with some cross-bred of the formulae of Hardy-Littlewood-Ingham (1926) and of Ingham (1926), arXiv: 2501.03646v1

    1926 arXiv

  4. [12]

    Moser, Jacob’s ladders, next equivalents of the Fermat-Wiles theorem and new infinite sets of equivalents generated by the Dirichlet’s series, arXiv: 2502.12748v1

    J. Moser, Jacob’s ladders, next equivalents of the Fermat-Wiles theorem and new infinite sets of equivalents generated by the Dirichlet’s series, arXiv: 2502.12748v1

    arXiv

  5. [13]

    J. Moser, Jacob’s ladders, new equivalents of the Fermat-Wiles theorem generated by certain cross-breed of Ingham and Heath-Brown formula (1979) and some chain of equivalents, arXiv: 2504.00479

    1979 arXiv

  6. [14]

    C. L. Siegel, ¨Uber Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Math., Astr. und Physik, Abt. B: Studien, 2, (1932), 45 – 80

    1932

  7. [15]

    Selberg, Contributions to the theory of the Riemann zeta-function, Arch

    A. Selberg, Contributions to the theory of the Riemann zeta-function, Arch. Math. og Naturv., B, pp. 89 – 155, (1946)

    1946

  8. [16]

    E. C. Titschmarsh, On van der Corput’s method and the zeta-function of Riemann(IV), Quart. J. Math. 5, (1934), 98 – 105. Department of Mathematical Analysis and Numerical Mathematics, Comenius Uni- versity, Mlynska Dolina M105, 842 48 Bratislava, SLOV AKIA Email address: jan.m...

    1934

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