Jacob's ladders and three new equivalents of the Fermat-Wiles theorem and an infinite set of these equivalents that are independent on the Jacob's ladders

Jan Moser

REVIEW 3 major objections 6 minor 1 cited by

Jacob's ladders and three new equivalents of the Fermat-Wiles theorem and an infinite set of these equivalents that are independent on the Jacob's ladders

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Every absolutely convergent Dirichlet series yields an analytic condition equivalent to the Fermat-Wiles theorem.

desk verdict The claimed new equivalents of FLT are tautological: Lemma 13 already shows the D-condition is just the Fermat inequality under a rescaling. read the letter →

arxiv 2412.12692 v1 pith:LNYZ6FSC submitted 2024-12-17 math.NT

Jan Moser This is my paper · ORCID

classification math.NT MSC 11M0611D41

keywords Fermat-WilestheoremRiemannzeta-functionDirichletseriesmean-valueSelbergintegralJacob'sladdersFermatrationalsequation

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 show that the classical mean-value theorem for Dirichlet series creates an infinite set of analytic conditions equivalent to the Fermat-Wiles theorem. For any absolutely convergent series $f(s)=\sum a_n n^{-s}$ at $\sigma_0$, with $F(\sigma_0;f)=\sum |a_n|^2 n^{-2\sigma_0}$, the normalized limit $(1/\tau)\int_0^{r\tau/F}|f(\sigma_0+it)|^2\,dt$ is shown to tend to $r$ for every $r>0$. Taking $r=(x^n+y^n)/z^n$ with $n\ge 3$, the condition that this limit is not $1$ is equivalent to excluding integer solutions of $x^n+y^n=z^n$. The first three equivalents use the zeta function, the $S_1(t)$ function, and a combination of both, and the proof of these is independent of Jacob's ladders; the final D-condition extends the result to every absolutely convergent Dirichlet series.

What carries the argument

The load-bearing object is the mean-value theorem for Dirichlet series (1.24), together with the associated constant $F(\sigma_0;f)$. This theorem converts the asymptotic mean square $\int_0^T |f(\sigma_0+it)|^2\,dt = FT + o(T)$ into an exact normalized limit by the substitution $T=x\tau/F$. The classical formula for the $2l$-th moment of $S_1(t)$ plays the same role for the second and third equivalents. Jacob's ladders, the paper's name for a family of near-linear functions $\varphi_1(T)$ and their reverse iterations $\varphi_1^{-k}(T)$ attached to integrals of $|\zeta(1/2+it)|^2$, appear only in the ladder-dependent equivalents; the main D-condition does not require them.

What would settle it

Take $f(s)=\zeta(s)$ at $\sigma_0=2$, so $F(2;\zeta)=\zeta(4)=\pi^4/90$, and take $r=(2^3+3^3)/4^3=35/64$. The paper predicts $\lim_{\tau\to\infty}\frac{1}{\tau}\int_0^{rF\tau}|\zeta(2+it)|^2\,dt = r$. If direct evaluation of this integral for increasing $\tau$ gave any other limit, the mean-value theorem behind the equivalence would be contradicted.

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Extended reading notes

Core claim

The central discovery is the limit identity (5.3): for every $f(s)=\sum a_n n^{-s}$ absolutely convergent at $\sigma_0$, and every $x>0$, \[\lim_{\tau\to\infty}\frac{1}{\tau}\$int_0^{{x\tau/F(\sigma_0;f)}}$|f(\sigma_0+it)|^2\,dt = x,\] where $F(\sigma_0;f)=\sum |a_n|^2 n^{-2\sigma_0}$. This is a scaling property obtained by substituting $T=x\tau/F$ into the classical mean-value theorem. When $x$ is replaced by a Fermat rational $r=(x^n+y^n)/z^n$ with $n\ge 3$, the same limit equals $r$, so the D-condition that the limit is not $1$ holds on the class of all Fermat rationals exactly when no solution to $x^n+y^n=z^n$ exists. The paper presents this as an infinite family of points of contact between the set of all Dirichlet series and the Fermat-Wiles theorem, with the proof independent of Jacob's ladders.

Load-bearing premise

The argument relies on the classical mean-value theorem for Dirichlet series, namely that the time average of $|f(\sigma_0+it)|^2$ over a long interval tends to the sum of the squared coefficients; if that limit failed for some $f$, the normalized limit would not equal the Fermat rational and the equivalence would collapse.

Editorial extensions

If this is right

For every fixed $\sigma>1$, the $\zeta$-condition (3.9) is a Fermat-Wiles equivalent using only the classical mean square of $\zeta$ on the line $\sigma$.
For every fixed $l\in\mathbb{N}$, the $S_1$-condition (3.16) gives an equivalent based on the mean square of $|S_1(t)|^{2l}$.
The combined condition (3.23) mixes the $\zeta$ and $S_1$ integrals and is again independent of Jacob's ladders.
The D-condition (1.29) yields an infinite set of equivalents, one for each absolutely convergent Dirichlet series.
Because every normalized limit explicitly equals the Fermat rational, each 'not equal to 1' condition is literally the statement that no Fermat solution exists for $n\ge 3$.

Reading between the lines

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

Editorial inference: The proof uses only the existence of an asymptotic mean-square law, so the same construction would produce a Fermat-Wiles condition for any function whose integral over $[0,T]$ is $cT+o(T)$, not just for Dirichlet series.
Editorial inference: These are logical equivalences rather than computational shortcuts; checking the D-condition for all Fermat rationals is the same task as proving the Fermat-Wiles theorem.
Editorial inference: The normalized-limit identity (5.3) is itself a general scaling law for mean-square integrals, so the method could attach similar statements to other known mean-value constants in analytic number theory.

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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,

Short version: the paper derives a family of limit statements that are all literally the Fermat inequality in disguise. The one genuinely transparent step — Lemma 13 — is also the step that sinks the claim. For every Dirichlet series f, the limit in (5.4) equals (x^n+y^n)/z^n, so the D-condition (5.5) is exactly q ≠ 1 with q = (x^n+y^n)/z^n. Any positive q would do; the specific features of f and σ0 cancel out. This is not a point of contact between Dirichlet series and FLT; it is a restatement of FLT after a linear rescaling of the integration variable.

What the paper does well: the substitutions are legal, and the author is unusually explicit about them. Lemma 2 and Lemma 13 state precisely what the limits are, so a reader can see the collapse without digging. The earlier zeta-function and Selberg sections follow the same pattern, and Section 4's Jacob-ladder version is too notation-dependent to verify from this text alone — the constant c is never defined here, and the reverse iterations are asserted rather than derived. The "infinite set" of equivalents is one condition written once per f, with no independence between them.

The overclaim is the problem, not the mathematics. Remark 2's suggestion that Hardy, Littlewood, and Selberg missed a deep connection is unsupported; this is a routine substitution in a mean value theorem. The references are almost entirely self-citations, and there is no engagement with any critical perspective on this kind of equivalence.

Who this is for: someone cataloguing ways to rewrite FLT as an analytic-looking inequality might want it for completeness, but a serious number theory reader will learn nothing. I would not bring it to our reading group, and I would not cite it. It deserves a desk reject with a short explanation, not referee time.

My recommendation: do not send this to peer review. If you want to be generous, a polite note pointing out that Lemma 13 makes the main theorem tautological would be enough.

Referee Report

3 major / 6 minor

Summary. The manuscript defines, for each Dirichlet series f absolutely convergent at σ0, the mean value F(σ0;f)=Σ|a_n|^2 n^{-2σ0}, and uses the classical mean-value theorem to show that lim_{τ→∞} (1/τ)∫_0^{(x/F(σ0;f))τ} |f(σ0+it)|^2 dt = x. It then substitutes x=(x^n+y^n)/z^n and asserts that the condition that this limit is different from 1 is a new 'D-equivalent' of the Fermat-Wiles theorem. Similar constructions are given for |ζ(σ+it)|^2, Selberg's |S1(t)|^{2l}, and linear combinations, with an additional set of examples involving Jacob's ladders in Section 4. The abstract claims an infinite set of points of contact between the set of all Dirichlet series and the Fermat-Wiles theorem, independent of Jacob's ladders.

Significance. If the claimed equivalences carried substantial analytic content, they would connect analytic number theory to a central Diophantine statement. However, the paper's own Lemmas show that each limit is exactly the Fermat rational, so the asserted 'equivalents' are logical restatements of the Fermat inequality rather than new mathematical contacts. The formal substitutions are correct, and the paper should be credited for stating clearly, in Lemmas 12–13, how the mean-value theorem forces the limit to equal the chosen x; but that clarity also exposes the circularity. The paper does not provide reproducible proofs beyond textbook mean-value calculations, and the claimed significance is not supported.

major comments (3)

[§5, Lemma 13, Eq. (5.4) and Theorem 7, Eq. (5.5)] comment
[§3, Theorems 1–3] comment
[§4, Eqs. (2.1), (2.2), (4.3), (4.16), (4.20)] comment

minor comments (6)

[§3, Lemma 1, Eq. (3.3)] comment
[§5, Lemma 12, Eq. (5.3)] comment
[§3, Theorem 2, Eq. (3.16)] comment
[§3, Eq. (3.18)] comment
[References] comment
[§1, Remark 2] comment

Circularity Check

1 steps flagged · score 8.0 of 10

The D-condition in Theorem 7 is constructed so that the limit equals the Fermat rational; the 'new equivalent' is just the Fermat inequality written in limit notation.

self definitional [Section 5, Eqs. (5.2)–(5.5), Lemmas 12–13 and Theorem 7; cf. Section 1.5, Eqs. (1.28)–(1.29).]
"if we put T = x/F(σ0; f) τ, x > 0 into (5.1), then ... Lemma 13. lim_{τ→∞} 1/τ ∫_0^{(x^n+y^n)/z^n τ / F(σ0;f)} |f(σ0+it)|^2dt = (x^n+y^n)/z^n ... Theorem 7. The D-condition lim_{τ→∞} 1/τ ∫_0^{(x^n+y^n)/z^n τ / F(σ0;f)} |f(σ0+it)|^2dt ≠ 1 ... represents new D-equivalent of the Fermat-Wiles theorem."

By (1.24), the integral over [0,T] is asymptotically F(σ0;f)T. The endpoint in Lemma 13 is deliberately chosen as (q/F)τ, so the coefficient F cancels after division by τ and the limit is q=(x^n+y^n)/z^n. Therefore the D-condition in Theorem 7 is exactly q≠1, i.e. x^n+y^n≠z^n, which is the Fermat-Wiles statement itself. No property of f beyond the mean-value theorem is used; replacing q by any positive number gives the condition q≠1. The infinite family of 'equivalents' is one inequality expressed with different normalizations, not an independent point of contact.

full rationale

The core derivation is mathematically correct but definitionally empty. From the classical mean-value theorem (1.24) the paper obtains (5.1), then substitutes T=x/F(σ0;f)τ, making the normalized integral tend to x by cancellation of F. Specializing x to the Fermat rational yields Lemma 13: the limit is (x^n+y^n)/z^n. Consequently the 'new D-equivalent' in Theorem 7, namely 'the limit is not 1', is logically identical to (x^n+y^n)/z^n≠1, i.e. to the negation of the Fermat equality. The same construction is repeated in Theorems 1–6 (ζ, Selberg, Jacob's-ladder integrals): in each case the upper limit is scaled by the reciprocal of the asymptotic coefficient so that the limit equals the Fermat rational, and the condition '≠1' is the Fermat inequality itself. No load-bearing self-citation occurs in the main Section 5 result, which rests only on (1.24); the circularity is definitional. Thus the claimed infinite set of points of contact reduces by construction to a single input inequality, so the circularity score is 8.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central Section 3 and 5 results rest only on classical mean value theorems, which are standard. Section 4 relies on the author's Jacob's ladder machinery, but the paper's advertised main result is independent of it. No new mathematical entities are posited.

assumptions (4)

standard math Classical mean value theorem for Dirichlet series: for absolutely convergent f(s) = Σ a_n n^{-s} at σ0, lim_{T→∞} (1/T) ∫_0^T |f(σ0+it)|^2 dt = Σ |a_n|^2 n^{-2σ0}.
Invoked in (1.24) and Section 5; it is the entire basis for the D-equivalent (5.5).
standard math Mean value theorem for ζ: ∫_1^T |ζ(σ+it)|^2 dt = ζ(2σ)T + o(T) for σ>1.
Used in Section 3, Lemma 1 (3.3).
standard math Selberg's mean value formula: ∫_0^T |S_1(t)|^{2l} dt = ¯c(l) T + O(T / log T).
Used in Lemma 3 (3.14) and Theorem 2.
domain assumption The Jacob's ladder constant c and reverse iterations φ_1^{-r} are well-defined and satisfy the properties in (2.1)-(2.14).
These are taken from the author's prior papers [3]-[7] and are not defined in this text; Section 4 depends on them.

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Cite this review

Pith. "Pith review of Jacob's ladders and three new equivalents of the Fermat-Wiles theorem and an infinite set of these equivalents that are independent on the Jacob's ladders." pith.science (2026). https://pith.science/paper/LNYZ6FSC

@misc{pith2026241212692,
  author       = {Pith},
  title        = {Pith review of: Jacob's ladders and three new equivalents of the Fermat-Wiles theorem and an infinite set of these equivalents that are independent on the Jacob's ladders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LNYZ6FSC}},
  note         = {Machine review of arXiv:2412.12692}
}

read the original abstract

In this paper we show that there is an infinite set of points of contact between the set of all Dirichlet's series and Fermat-Wiles theorem. The proof is independent on the Jacob's ladders.

Discussion (0). Continue with ORCID to comment.

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

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 · 8 canonical work pages · cited by 1 Pith paper

[1]

Hardy, J.E

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arXiv 2010
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arXiv 2011
[5]

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arXiv
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1930

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [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), 1 19 – 196, (1918)

work page 1918

[2] [2]

L. D. Kudryavtsev, Mathematical Analysis , vol. 1., Nauka: Moscow, 1973

work page 1973

[3] [3]

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

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

arXiv 2010

[4] [4]

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

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

arXiv 2011

[5] [5]

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

arXiv

[6] [6]

Moser, Jacob’s ladders, interactions between ζ-oscillating systems and ζ-analogue of an elementary trigonometric identity, Proc

J. Moser, Jacob’s ladders, interactions between ζ-oscillating systems and ζ-analogue of an elementary trigonometric identity, Proc. Stek. Inst. 299, 189 – 204, (2017)

work page 2017

[7] [7]

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 14 JAN MOSER

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 14 JAN MOSER

arXiv

[8] [8]

Moser, Jacob’s ladders, existence of almost linear in crements of the Hardy-Littlewood integral and new types of multiplicative laws, arXiv: 2304

J. Moser, Jacob’s ladders, existence of almost linear in crements of the Hardy-Littlewood integral and new types of multiplicative laws, arXiv: 2304. 09267

work page