---

Claim. D. Duverney, Irrationalité d'un q-analogue de ζ(2), C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 10, 1287–1289. The note's Théorème (p. 1287, proof p. 1289) states that for every integer $q$ other than $-1$, $0$ and $1$ the number

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

is irrational. At $q=2$ the factor $(q-1)^2$ is $1$ and $\zeta(2;2)$ is the series $\sum_{n\ge1}\sigma(n)/2^n$ of Problem 250, so the answer to the question is yes; for every other integer base the same theorem settles the all-base form in which Erdős first posed the question. The proof is elementary: Euler's pentagonal number theorem, the note's Lemme that $1$, $f(1/q)$ and $(1/q)f'(1/q)$ are linearly independent over $\mathbb{Q}$ for $f(x)=\prod_{n\ge1}(1-x^n)$, and the logarithmic derivative $xf'(x)/f(x)=-\sum_{n\ge1}nx^n/(1-x^n)$. The result page Théorème and the source card hold the statement, the reconstruction and the records. The note cites Erdős's 1948 and 1988 statements of the question and answers only the irrationality asked; transcendence is Nesterenko's later and stronger result, on [[problems/irrationality/E0250/claims/1996_03_07_nesterenko|its own claim page]].

Acceptance. Refereed: Comptes Rendus de l'Académie des Sciences, Série I, volume 321 (1995); the note was received on 18 September 1995 and accepted after revision on 25 September 1995, presented by Jean-Pierre Serre. Reviewed: the zbMATH review Zbl 0843.11034 (reviewer J. Hančl) records the theorem without objection, and later refereed papers on the irrationality measure of $\zeta_q(2)$ (Zudilin 2002; Postelmans and Van Assche 2007; Smet and Van Assche 2009) treat the irrationality as settled. The site's remarks credit only Nesterenko; a comment of 2026-09-05 in the site's discussion thread raised this earlier reference, and the page was unchanged at the snapshot of 2026-09-17. Separately from this acceptance, the library's complete reconstruction of the Lemme and the Théorème was independently reviewed (fresh-context review, verdict refutation-failed, and a distinct passing grade, under the card's evidence/verify/) relative to Euler's theorem and to Théorème 2 of Duverney 1993; that review is compilation proof coverage, not the acceptance evidence listed above.

Depends on. Nothing in this wiki; the claim is the cited note's theorem.

The page name carries the date printed on the note as its date of receipt; the issue's day of publication is not stated in the records the library holds.