---

Refutation-failed for the complete rewritten proofs of the Lemme and of the Théorème of Duverney 1995, relative to Euler's pentagonal number theorem (E), consumed as an external statement whose proof was not inspected, and to Théorème 2 of Duverney 1993 (T2), whose statement and half-page proof were both checked against the 1993 paper. Fresh-context reviewer, Claude Fable 5.1, under a refutation charge; dated 2026-09-17. This report records no grade; the distinct grader's record is filed separately beside it.

Subject and independence

Frozen subject. Two pages of this card and one page of the 1993 card, read whole, with the sections named below as the reviewed mathematics:

1. library/irrationality/duverney_1995_irrationalite_q_analogue_zeta_2/lemme.md (Lemme): Statement, Premises, Complete rewritten proof (Steps 0--7), Remarks.
2. library/irrationality/duverney_1995_irrationalite_q_analogue_zeta_2/theoreme.md (Théorème): Statement with its Specialization, Premises, Complete rewritten proof (Steps 1--4), Remarks.
3. library/irrationality/duverney_1993_proprietes_arithmetiques_serie_fonctions_theta/theoreme_2.md (Théorème 2): the Statement, as the exact external criterion consumed, and its proof sketch.

Consequence sentences read against the verdict: the Compiled scope section of this card's source card; the Compiled proof coverage paragraph of Problem 250 (the page was read whole); the Use in Duverney 1995 and Coverage sections of the Théorème 2 page. The Verification sections of the two subject pages and the updated fields are standing wording, outside the mathematical subject.

Exposure and materiality. The reviewer's whole-page reading included standing text outside the mathematical subject: the Verification sections of the retained copies evidence/assets/reviewed_pages/lemme.md (lines 203-214) and evidence/assets/reviewed_pages/theoreme.md (lines 160-174), the Coverage section of evidence/assets/reviewed_pages/duverney_1993_theoreme_2.md (lines 76-79), the status field and Status paragraph of wiki/problems/irrationality/E0250/_index.md (lines 7 and 24 as they stood at 2026-09-17T07:01:06Z, the expanded paragraph of the working tree landed at 2026-09-17T09:49:34Z), the card's pre-landing Compiled scope section (bytes not retained) and the author's summary named above (not retained); a separately spawned materiality grader, Claude Fable 5.1, ruled on 2026-09-18 by the content test that this exposure is immaterial, because none of that text states or implies whether the reconstructions are faithful and complete and the verdict rests on the rederivations from the page images.

Revision. At review time (2026-09-17) the three pages were uncommitted files in the working tree, whose committed state was that of 2026-09-17T07:01:06Z; the review precedes the filing that lands them, and that filing is the one that first adds each page. That landing filing is the one of 2026-09-17T09:49:34Z, which first adds the three reviewed pages with their retained snapshots. Because the reviewed bytes were not yet committed, byte-identical copies are retained as opaque attachments under evidence/assets/reviewed_pages/ of this card (lemme.md, theoreme.md, duverney_1993_theoreme_2.md); they are the reviewed bytes and are not edited. Their relation to the current pages: identical at review time; the subject pages' Verification sections are expected to change at filing, and diff against the copies afterwards separates later edits from the reviewed text.

Sources read. Both PDFs are the cards' folder-name files; their sizes and SHA-256 values were recomputed and agreed with the provenance lines the two cards then carried, and each is identified by its path and its Git LFS pointer. Neither has a text layer; every page was rendered with pdftoppm -r 200 -png and read as an image.

• Duverney 1995, printed pp. 1287--1289 (PDF pp. 1--3), read in full: (1)--(5) and the Théorème, (6)--(13) and the Lemme with its proof, (14)--(15), the closing sentence, the dates and the eight references.
• Duverney 1993, printed p. 175 (PDF p. 1, title and identity), p. 176 (PDF p. 2, Théorème 2), p. 178 (PDF p. 4, section 2, the proof and the start of the Remarque) and p. 179 (PDF p. 5, the end of the Remarque), read in full. Théorème 1, sections 3--5 and pp. 180--188 were not read.

Allowed operating reading. The repository instructions, the guidance pages on anatomy, evidence, verification, tools and mathematical authoring, the commission's ground rules and this review's assignment. The commission text included the reconstruction author's report, a step-by-step summary of the pages and of the author's own checks; that is the only exposure to author material beyond the pages, it contains no argument absent from the pages, and every premise and deduction below was derived from the pages and the PDFs, not from that summary. Excluded and not read: private research notes and plans and their mathematical paraphrases, other reviews' reports and verdicts, the author's private scratch computations, and the note's cited books.

Independence. The reviewer did not author, edit or build on the reconstruction or on the 1993 premise page before this review and worked in a context that held none of them. No collaborator of the author took part. No computation is part of the argument or of this verdict; the reviewer's finite checks of the printed identities (below, "Sanity aids") are working-storage aids, not filed evidence.

Restatement

Throughout, $q\in\mathbb Z$ with $|q|\ge2$, that is $q\in\mathbb Z\setminus {-1,0,1}$, and $f(x)=\prod_{n\ge1}(1-x^n)$ for real $|x|<1$.

Lemme. For every such $q$ there is no triple $(c_0,c_1,c_2)\in\mathbb Q^3\setminus{0}$ with $c_0+c_1f(1/q)+c_2\,(1/q)f'(1/q)=0$. Equivalently: for every pair of integers $(a,b)\ne(0,0)$ the real number $\alpha_q=a\,f(1/q)+b\,(1/q)f'(1/q)$ is irrational.

Théorème. For every such $q$ the number $\zeta(q;2)=\sum_{n\ge1}q^n \big((q-1)/(q^n-1)\big)^2$ is irrational. The page also asserts, and proves, that $\zeta(q;2)=(q-1)^2\sum_{n\ge1}n/(q^n-1)=(q-1)^2\sum_{n\ge1} \sigma(n)/q^n$, with $\sigma(n)=\sum_{d\mid n}d$, and draws the specialization: $\sum_{n\ge1}\sigma(n)/2^n$ is irrational, and $\sum_{n\ge1}\sigma(n)/q^n$ is irrational for every integer $|q|\ge2$.

Statement fidelity against the page images

• Théorème (p. 1287): "Si $q\in\mathbb Z-\{-1,0,1\}$, $\zeta(q;2)$ est irrationnel", with (1) defined for $q\in\mathbb C$, $|q|>1$. The page's Statement is this, with (3) and (5) as printed on p. 1287 and $d_1=\sigma$ as in (4). Faithful.
• Lemme (p. 1288): "Si $q\in\mathbb Z-\{-1,0,1\}$, les nombres $1$, $f(1/q)$ et $(1/q)f'(1/q)$ sont linéairement indépendants sur $\mathbb Q$", with $f$ defined by (6) for $|x|<1$. Faithful.
• (6), (7), (8), (9), (10), (11), (12), (13), (14), (15): each display on the pages agrees with the print, including the exponents $n(3n\pm1)/2$, the signs $(-1)^n$, the constant term $a$ in (9), the range $n\ge0$ in (10) and (12), the sign in (14) and the denominators $q^n-1$ in (15).
• The note's citations are reproduced correctly: (E) to "[2], p. 124; [6], p. 229"; T2 to "le théorème 2 de [3]"; (5) to "[8], p. 257"; the route of the deduction to Bundschuh and Väänänen [1].
• Two slips of the print are correctly reported by the Lemme page: the note writes "si $x_q=\eta/\delta$" before (12) where (10) names the number $\alpha_q$ ($x_q$ is the 1993 paper's notation), and (11) records $k$ zeros after $n_k$ where the exponent pattern gives $2k$; T2 needs only $k$.
• T2 (Duverney 1993, p. 176): the Théorème 2 page's Statement reproduces hypotheses (a), (b) with (b$_1$), (b$_2$), (c) with (c$_1$), (c$_2$), the series $x=\sum_{n\ge0}a(n)q^{-n}$ and conclusion (4) exactly, including the quantifier "il existe une infinité d'entiers $k$" and the conclusion "pour $k$ assez grand". Its proof sketch matches section 2 on p. 178 step for step, and its summary of the Remarque matches pp. 178--179.
• The Théorème page's Specialization says Erdős posed the all-base form in 1948 and 1957. The note itself cites the 1948 paper and the 1988 survey; the 1957 attribution rests on a paper outside this review's sources and is consistent with the Origin section of the Problem 250 page. It is a historical remark, not a deduction, and was not checked further.

Essential deductions, rederived

The Lemme (Steps 0--7 of the page)

Step 0, reduction. A nontrivial rational relation among $1$, $f(1/q)$, $(1/q)f'(1/q)$ clears to integers $(c_0,c_1,c_2)\ne0$; $c_1=c_2=0$ would force $c_0=0$, so $(c_1,c_2)\ne(0,0)$ and $c_1f(1/q)+c_2(1/q)f'(1/q)=-c_0 \in\mathbb Q$. Hence the irrationality of every $\alpha_q$ with $(a,b)\ne(0,0)$ implies the Lemme. Correct; this is the note's "il suffit".

Step 1, (9). By (E) $f$ equals on $(-1,1)$ the power series $\sum_{m\ge0}c_mx^m$ with $c_m\in\{0,\pm1\}$, whose radius of convergence is at least $1$; a power series is differentiable inside its disk with termwise derivative, so $xf'(x)=\sum_m mc_mx^m$, which is (8). Evaluating (7) and (8) at $x=1/q\in(-1,1)\setminus\{0\}$ gives (9). Each series in (9) converges absolutely: the coefficient is $O(n^2)$ and $n(3n-1)/2\ge n$ for $n\ge1$, so the terms are $O(n^2\,2^{-n})$. Correct.

Step 2, (10). With $p_m^-=m(3m-1)/2$ and $p_m^+=m(3m+1)/2$ one has $p_m^+-p_m^-=m$ and $p_{m+1}^--p_m^+=\big((3m^2+5m+2)-(3m^2+m)\big)/2=2m+1$, so $1=p_1^-<p_1^+<p_2^-<\cdots$ and all exponents in (9) are distinct positive integers (each $m(3m\pm1)$ is even). Setting $a(0)=a$, $a(p_m^\pm)=(-1)^m(a+bp_m^\pm)$ and $a(n)=0$ elsewhere gives an integer sequence, and $\sum_{n\ge0}a(n)q^{-n}$ is the absolutely convergent series (9) with its terms reordered and zeros inserted, hence equals $\alpha_q$. Correct.

Step 3, zero runs. After $n_k=p_k^+$ the next exponent is $p_{k+1}^-=n_k+2k+1$, so $a(n_k+j)=0$ for $1\le j\le2k$ (Z1), which contains (11). Before $n_k$ the previous exponent is $p_k^-=n_k-k$, so $a(n_k-j)=0$ for $1\le j\le k-1$ (Z2), empty for $k=1$; $a(n_k-k)= (-1)^k(a+bp_k^-)$ is in general nonzero. Correct, and exactly the two runs the note uses.

Step 4, hypotheses of T2 with $r(n)=n^2$. (a): $a(n_k)=(-1)^k(a+bn_k)$ vanishes for at most one $k$ when $b\ne0$ (strict monotonicity of $n_k$) and never when $b=0\ne a$. (b): for $n\ge c=|a|+|b|\ge1$, a nonzero $a(n)$ equals $\pm(a+bn)$, so $|a(n)|\le|a|+|b|n\le cn\le n^2$. (b$_1$): $n^2>0$ for $n\ge1$; the theorem's (b) is stated for $n$ large, and the proof evaluates $r$ only at $n\ge n_k+k+1$ and in the ratio (b$_2$), so $r(0)=0$ is immaterial (one may also take $r(n)=\max(1,n^2)$ with no other change). (b$_2$): $(1+1/n)^2\to1<2\le|q|$. (c): all $k\ge1$ with $n_k=k(3k+1)/2$; (c$_1$) is (Z1); (c$_2$): $n_k+k+1=(3k^2+3k+2)/2\le4k^2$ since $5k^2-3k-2=(5k+2)(k-1)\ge0$, so $r(n_k+k+1)/|q|^k\le16k^4/2^k\to0$. Correct.

Step 5, (12). T2 applied to $x=\alpha_q=\eta/\delta$ gives $k_0$ with $\eta q^{n_k}-\delta\sum_{n=0}^{n_k}a(n)q^{n_k-n}=0$ for all $k\ge k_0$. Correct.

Step 6, $q^k\mid\delta a(n_k)$. Split the sum at $n_k-k\ge1$: the indices $n_k-k+1,\dots,n_k-1$ contribute nothing by (Z2); every index $n\le n_k-k$ carries $q^{n_k-n}$ with $n_k-n\ge k$; and $n_k\ge k$. So $\delta a(n_k)=\eta q^{n_k}-\delta\sum_{n\le n_k-k}a(n)q^{n_k-n}$ is an integer multiple of $q^k$. The term at $n=n_k-k$, whose coefficient $a(p_k^-)$ is in general nonzero, carries exactly $q^k$; the accounting is tight and correct. The sign of $q$ plays no role.

Step 7, growth and contradiction. By (13), $|\delta a(n_k)|\le |\delta|(|a|+|b|)n_k\le2|\delta|(|a|+|b|)k^2$ since $n_k=(3k^2+k)/2\le 2k^2$. A nonzero multiple of $q^k$ has absolute value at least $2^k$, which exceeds $2|\delta|(|a|+|b|)k^2$ for large $k$; so $\delta a(n_k)=0$, hence $a+bn_k=0$, for all large $k$. Two values $k<k'$ give $b(n_{k'}-n_k)=0$, so $b=0$ and then $a=0$, against $(a,b)\ne(0,0)$. Correct. This is the note's "ceci est impossible".

The Théorème (Steps 1--4 of the page)

Step 1, (1) = (3) = (5). With $t=q^{-n}$, $0<|t|\le1/2$, $q^n((q-1)/(q^n-1))^2=(q-1)^2t/(1-t)^2=(q-1)^2\sum_{j\ge1}jt^j$. The double series $\sum_n\sum_jj\,q^{-nj}$ converges absolutely since $\sum_j j|q|^{-nj}=|q|^{-n}/(1-|q|^{-n})^2\le4|q|^{-n}$; summing over $n$ first gives $\sum_j j/(q^j-1)$, which is (3); grouping by $m=nj$ gives $\sum_m\sigma(m)q^{-m}$, which is (5). Correct. Write $D_q=\sum_jj/(q^j-1)$.

Step 2, the product, $f(1/q)>0$ and (14). For $x\in[-\rho,\rho]$, $0<\rho<1$: $1-x^n\in[1-\rho^n,1+\rho^n]$, so $\ell_n(x)=\log(1-x^n)$ satisfies $|\ell_n(x)|\le\max(\log(1+\rho^n),-\log(1-\rho^n))= -\log(1-\rho^n)\le\rho^n/(1-\rho)$, using $-\log(1-t)\le t/(1-t)$; the M-test gives uniform convergence of $\sum\ell_n$ to $g$, and the partial products $\exp(\sum_{n\le N}\ell_n)$ converge to $f=e^g>0$. The derivative series $\sum\ell_n'$, $\ell_n'(x)=-nx^{n-1}/(1-x^n)$, is dominated by $\sum n\rho^{n-1}/(1-\rho)<\infty$, so $g'=\sum\ell_n'$ on $(-\rho,\rho)$ and, $\rho$ being arbitrary, $f'=g'f$ on $(-1,1)$, which is (14). The function $f$ is the same one that (E) identifies with the series (7), so this $f'$ is the Lemme's $f'$. Correct.

Step 3, (15). At $x=1/q$, division by $f(1/q)>0$ gives $(1/q)f'(1/q)/f(1/q)=-\sum_nnq^{-n}/(1-q^{-n})=-\sum_nn/(q^n-1)=-D_q$. Correct; the note does not remark on $f(1/q)\ne0$, and the page supplies it.

Step 4, conclusion. If $\zeta(q;2)\in\mathbb Q$ then $D_q=\zeta(q;2)/(q-1)^2\in\mathbb Q$, and (15) reads $0\cdot1+D_q\cdot f(1/q)+1\cdot(1/q)f'(1/q)=0$, a rational relation with a nonzero coefficient, against the Lemme. So $\zeta(q;2)\notin\mathbb Q$, and $\sum\sigma(n)/q^n=\zeta(q;2)/(q-1)^2\notin\mathbb Q$ because $(q-1)^2$ is a nonzero integer; at $q=2$ the factor is $1$. Correct. (The Théorème uses from the Lemme only that $f(1/q)$ and $(1/q)f'(1/q)$ are $\mathbb Q$-linearly independent; the Lemme gives more.)

The external criterion T2 (Duverney 1993, p. 178), checked

If $\beta x=\alpha$ then $\alpha q^{n_k}-\beta\sum_{n\le n_k}a(n)q^{n_k-n} =\beta q^{n_k}\sum_{n>n_k}a(n)q^{-n}=\beta q^{n_k}\sum_{n\ge n_k+k+1} a(n)q^{-n}$ by (c$_1$). For $k$ large, $n_k+k+1\ge k+1$ exceeds the thresholds of (b) and of a ratio bound $r(n+1)/r(n)\le\eta<|q|$ from (b$_2$), so $|a(n)|\le r(n)\le r(n_k+k+1)\eta^{\,n-(n_k+k+1)}$ for $n\ge n_k+k+1$, and the geometric sum bounds the left side by $|\beta|\,r(n_k+k+1)/\big(|q|^k(|q|-\eta)\big)$, which tends to $0$ by (c$_2$). The left side is an integer, so it vanishes for $k$ large. Correct. The paper's remark that $n_k\to\infty$ "en vertu de (a)" is true (the premise page's added justification is valid: a bounded subsequence of $n_k$ with (c$_1$) would annihilate $a(n)$ for all large $n$) but not needed, since $n_k+k+1\ge k+1$ already passes the thresholds.

External premises

No native L-claim is consumed. There is no batch and no acceptance order.

• (E) Euler's pentagonal number theorem, as printed in (7) of the note for real $|x|<1$: $\prod_{n\ge1}(1-x^n)=1+\sum_{n\ge1}(-1)^n \big(x^{n(3n+1)/2}+x^{n(3n-1)/2}\big)$. Interface: used once, in Step 1 of the Lemme, to identify $f$ on $(-1,1)$ with a power series; the Théorème page uses it only to identify its product-defined $f'$ with the Lemme's. Reading depth: claims checked against the print and against the classical identity known to the reviewer, with Franklin's involution and the Jacobi triple product as its standard proofs; no proof inspected in this review and the note's two cited books not consulted. Basis for reliance: a classical theorem of the literature, cited by the note to two standard references. It remains an external premise of both proofs.
• (T2) Théorème 2 of Duverney 1993, Acta Arith. 64 (1993), statement p. 176, proof p. 178, exactly as restated on the Théorème 2 page. Interface: applied with the present $q$, the sequence $a(n)$ of Step 2, $r(n)=n^2$ and $n_k=k(3k+1)/2$ for all $k\ge1$; the conclusion (4) is the note's (12). Reading depth in this review: proof verified, as recorded above. The remaining sections of the 1993 paper were not read and receive no coverage.
• Elementary analysis, used without citation and accepted: termwise differentiation of a power series inside its disk; invariance of the sum of an absolutely convergent series under rearrangement and insertion of zeros; summation of an absolutely convergent double series in any order; the M-test; differentiation of a convergent series of differentiable functions whose derivative series converges uniformly on an interval; continuity of $\exp$; and $-\log(1-t)\le t/(1-t)$ for $0\le t<1$.

Weakest steps, rederived

1. Step 6 of the Lemme. The divisibility needs exactly the $k-1$ zeros before $n_k$ and the exponent $k$ on the term at $n_k-k$; both come from the gap $p_k^+-p_k^-=k$, rederived in Step 2. Had the run been one shorter, only $q^{k-1}\mid\delta a(n_k)$ would follow, and Step 7 would still close with $2^{k-1}$; the argument has slack and the accounting on the page is exact.
2. Step 4 of the Lemme. The criterion is applied at its printed strength: (b) holds from $n=|a|+|b|$ on, (c) with the full sequence $k\ge1$, and the conclusion "for $k$ large" is then unconditional. The only possible misreading, (b$_1$) at $n=0$, is immaterial to the proof of T2 and removable by $r(n)=\max(1,n^2)$. T2's own proof was checked.
3. Steps 2--3 of the Théorème. The division in (15) needs $f(1/q)\ne0$, which the note does not state; the page proves $f>0$ on $(-1,1)$ through the uniformly convergent logarithmic series, and the same series justifies (14). Both bounds in the derivation were rederived above, including the two-sided bound on $\log(1-x^n)$ that the page leaves implicit.

Strongest attempted refutation

The reviewer tried to break the arithmetic contradiction rather than the analysis. (i) Choosing $(a,b)$ so that $a(n_k)=0$ for some $k$ does not help: at most one $k$ is affected, and Step 7 needs the vanishing for all large $k$. (ii) A negative $q$ changes no step: (E) holds for negative $x$, T2 is stated for $|q|\ge2$, divisibility is in $\mathbb Z$, and $f(1/q)>0$ because every factor $1-x^n$ is positive on $(-1,1)$. (iii) The printed slip "$x_q=\eta/\delta$" cannot redirect T2 to another number: (10) names $\alpha_q$ and (12) is T2's (4) for it. (iv) Shortening the zero run before $n_k$ is impossible: the previous pentagonal exponent is exactly $n_k-k$. (v) The exchanges of summation behind (3) and (5) and the termwise differentiation behind (8) and (14) are covered by absolute or uniform convergence at every $|q|\ge2$. (vi) The deduction of the Théorème requires $(q-1)^2\ne0$ and a relation with a nonzero coefficient; both hold. No attack produced a counterexample, an unsupported step or a misapplied premise.

Checklist

Consequence sentences

The card's Compiled scope, the Compiled proof coverage paragraph of Problem 250 and the Théorème 2 page's Use in Duverney 1995 describe the reconstruction correctly: complete rewritten proofs, (E) and T2 as external premises, author-recorded with review pending, and the problem's status resting on the refereed note and on Nesterenko's theorem independently of the reconstruction. After the distinct grade is filed, those sentences and the two Verification sections may say that the reconstruction was independently reviewed with verdict refutation-failed, relative to (E) as an unproved external premise and to T2 at proof verified depth; they must not say that (E) was proved here or that anything about Nesterenko's proof was reviewed.

Sanity aids

Not evidence and not filed: in working storage the reviewer checked, in exact integer arithmetic, that the coefficients of $\prod_{n\le4000} (1-x^n)$ agree with (7) through $x^{4000}$, that the gaps $m$ and $2m+1$ and the runs (Z1), (Z2) and the values (13) hold for $k\le250$ and five pairs $(a,b)$, and that $\lvert a(n)\rvert\le n^2$ from $n=\lvert a\rvert+ \lvert b\rvert$; and, to 60 decimal digits at $q\in\{2,-2,3,-3,7,-10\}$, that (7), (8), (9), (10), (14), (15) and (1) = (3) = (5) agree and that $f(1/q)>0$. These aids illustrate the printed identities; the verdict rests on the derivations above.

Verdict and scope

Refutation-failed. The Statement, Premises and Complete rewritten proof of the Lemme page and of the Théorème page are faithful to Duverney 1995 and complete: every essential deduction of the note's proofs, and every expansion the pages add, was rederived and found correct, relative to (E), an external statement not proved here, and to T2, whose statement and proof were checked against Duverney 1993. The Specialization to $q=2$ and to every integer base is a correct consequence.

Limitations. This review proves nothing about (E) beyond its identity with the classical theorem; it does not review Nesterenko's transcendence proof, the Bundschuh--Väänänen route, the Zbl or journal metadata on the card, or any other page of the two cards; it changes no problem status (Problem 250 was proved before and after, on the refereed publications) and confers no native tier, the subject being source results without an L-claim. The reconstruction becomes independently accepted compilation proof coverage only when the distinct grader's record passes this report under the contract; until then it remains author-recorded with this review attached.

Recommendations, none required for the verdict: (i) the Théorème page's Premises bullet could say that the Théorème uses only the $\mathbb Q$-linear independence of $f(1/q)$ and $(1/q)f'(1/q)$; (ii) the Lemme page's (b$_1$) remark could mention $r(n)=\max(1,n^2)$; (iii) the Specialization's "1948 and 1957" could point to the Origin section of Problem 250, where the 1957 paper is cited; (iv) the Théorème 2 page could note that $n_k\to\infty$ is not needed for the proof.