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Claim. W. Zudilin, On the irrationality measure for a q-analogue of ζ(2), Mat. Sb. 193 (2002), no. 8, 49--70; English translation Sb. Math. 193 (2002), no. 8, 1151--1172. The paper's Theorem (pp. 1151--1152 of the translation): for $q=1/p$ with $p\in\mathbb Z\setminus\{0,\pm1\}$, the number $\zeta_q(2)=\sum_{n\ge1}q^n/(1-q^n)^2$ is irrational, and its irrationality measure satisfies $\mu(\zeta_q(2))\le4.07869374\ldots$. At $p=2$, $\zeta_{1/2}(2)=\sum_{n\ge1}2^n/(2^n-1)^2=\sum_{n\ge1}\sigma(n)/2^n$, the series of Problem 250, so the theorem answers the question yes by a proof independent of Duverney's and Nesterenko's. The paper credits Duverney with the first proof of the irrationality. The page carries the date on which Mat. Sb. received the paper, 8 November 2001.

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

Acceptance. Refereed: Matematicheskii Sbornik, volume 193, with the English translation in Sbornik: Mathematics. The proof is recorded by statement and pointer only.