Duverney 1995 irrationalite q analogue zeta 2

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evidence/: Retains the independent review and distinct grade of the complete Lemme and Théorème reconstructions, and the earlier focused statement-fidelity record.

lemme: Proves, in a complete, independently reviewed reconstruction, that for the Euler product f of (1 - x^n) and any integer q other than -1, 0 and 1, the numbers 1, f(1/q) and (1/q) f'(1/q) are linearly independent over the rationals; the tool behind the Théorème.

theoreme: Proves, in a complete, independently reviewed reconstruction, that the q-analog zeta(q;2), equal to (q-1)^2 times the sum of sigma(n) over q^n, is irrational for every integer q other than -1, 0 and 1; at q = 2 this is the divisor-sum series of problem 250.

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D. Duverney, Irrationalité d'un q-analogue de ζ(2), C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 10, 1287--1289; Zbl 0843.11034 (reviewer J. Hančl). "Note remise le 18 septembre 1995, acceptée après révision le 25 septembre 1995", presented by Jean-Pierre Serre.

The copy read for this card is the author's scan of the three printed pages (running head "C. R. Acad. Sci. Paris, t. 321, Série I, p. 1287-1289, 1995"). It has no text layer; every statement and formula below was read on the page images. Provenance: fetched from https://danielduverney.fr/documents/theorie-des-nombres/qAnalogueZeta(2).pdf (listed at https://danielduverney.fr/mathematical-publications/) on 2026-09-17 (UTC), 2,438,038 bytes. The journal issue (t. 321, no. 10) is also online in Gallica (https://gallica.bnf.fr/ark:/12148/bpt6k6204514p, printed pp. 1287--1289 at views 19--21; record and page map read); no other version was compared. The scan prints "0764-4442/95/03211287 $ 2.00 © Académie des Sciences" at the foot of its first page (printed p. 1287, read on the page image), every other right reserved.

The transcriptions on this card and its result pages follow that scan.

Contents

For $q\in\mathbb C$ with $|q|>1$ the note defines

$$\zeta(q;2)=\sum_{n=1}^{\infty}q^n\Big(\frac{q-1}{q^n-1}\Big)^2 \qquad (1)$$

and notes $\lim_{q\to1}\zeta(q;2)=\zeta(2)$ (2). Expanding $(1-q^{-n})^{-2}=\sum_{k\ge0}(k+1)q^{-nk}$ and exchanging the summations gives (3), $\zeta(q;2)=(q-1)^2\sum_{n\ge1}n/(q^n-1)$; since (3) is a Lambert series (Hardy and Wright, p. 257), one has with $d_m(n)=\sum_{d\mid n}d^m$ (4) the form (5), $\zeta(q;2)=(q-1)^2\sum_{n\ge1}d_1(n)/q^n$, and $d_1=\sigma$. The note then says: "Sous la forme (5), le problème de l'irrationalité de $\zeta(q;2)$ a été posé, à plusieurs reprises, par Paul Erdös ([4], [5])" (p. 1287), where [4] is the 1948 Lambert-series paper (closing remark, p. 66) and [5] the 1988 Durham survey (p. 102).

• Théorème (p. 1287; proof p. 1289): if $q\in\mathbb Z\setminus\{-1,0,1\}$, then $\zeta(q;2)$ is irrational. At $q=2$ the factor $(q-1)^2$ is $1$ and the number is $\sum_{n\ge1}\sigma(n)/2^n$, the number of Problem 250.
• Lemme (p. 1288; proof pp. 1288--1289): for the same $q$ and $f(x)=\prod_{n\ge1}(1-x^n)$, the numbers $1$, $f(1/q)$ and $(1/q)f'(1/q)$ are linearly independent over $\mathbb Q$. Its tools are Euler's pentagonal number theorem, formulas (6)--(8), and Théorème 2 of the author's 1993 Acta Arithmetica paper, filed as theoreme_2.
• The Théorème follows from the Lemme through the logarithmic derivative (14), $xf'(x)/f(x)=-\sum_{n\ge1}nx^n/(1-x^n)$, its value (15) at $x=1/q$, and (3). The note says it follows the route used by Bundschuh and Väänänen (Arithmetical investigations of a certain infinite product, Compositio Math. 91 (1994), no. 2, 175--199; the note's reference [1] prints pp. 175--201) for $\sum_{n\ge1}1/(q^n-1)$.

The note's references: [1] Bundschuh and Väänänen 1994; [2] Chandrasekharan, Elliptic functions, 1985; [3] Duverney, Acta Arith. 64 (1993), 175--188; [4] Erdős, J. Indian Math. Soc. 12 (1948), 63--66; [5] Erdős, in New advances in transcendence theory, 1988; [6] Exton 1983; [7] Gasper and Rahman 1990; [8] Hardy and Wright, 5th edition, 1989.

The note concerns only the divisor-sum series. It says nothing about the totient series of Problem 249 or the prime-factor series of Problem 69, so no link to those pages is made.

Compiled scope

Both statements and every displayed formula were read on the page images. Complete rewritten proofs of the Lemme and of the Théorème are filed on their pages, with Euler's pentagonal number theorem and Théorème 2 of Duverney 1993 as identified external premises and the steps the note leaves to the reader (the nonvanishing $f(1/q)\ne0$, the ordering of the generalized pentagonal numbers behind (10) and (11), the divisibility and growth steps after (12)) written out. Both were independently reviewed: the fresh-context review under this card's evidence/verify/ reached the verdict refutation-failed, and the distinct grade passed it. They are independently accepted compilation proof coverage relative to Euler's theorem, consumed as an unproved external statement, and to Théorème 2, whose statement and proof were checked; nothing about Nesterenko's proof was reviewed. The status of Problem 250 rests on this refereed note together with Nesterenko's transcendence theorem (card), not on the local reconstruction. An earlier focused statement-fidelity review of the campaign's extraction pages of this note and of their exact Problem 250 specialization, filed as statement_fidelity, has verdict refutation-failed for that frozen extraction, disclosed proof outline and exact specialization; it awards no tier and does not extend the reconstruction review.

Bears on. #250: the Théorème at $q=2$ settles the exact question in the affirmative; it is the first published proof of that irrationality.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.