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Claim. C. Smet and W. Van Assche, Irrationality proof of a q-extension of ζ(2) using little q-Jacobi polynomials, Acta Arith. 138 (2009), no. 2, 165--178 (arXiv:0809.2501, first posted 15 September 2008, the date this page carries). Theorem 1.1 (p. 2 of the arXiv version): for $q=1/p$ with $p\in\{2,3,\dots\}$, the number $\zeta_q(2)=\sum_{k\ge1}kq^k/(1-q^k)$ is irrational, with irrationality measure $\mu(\zeta_q(2))\le10\pi^2/(5\pi^2-24)=3.8936\ldots$. At $p=2$ the number is $\sum_{n\ge1}\sigma(n)/2^n$, 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.

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

Acceptance. Refereed: Acta Arithmetica, volume 138. The proof is recorded by statement and pointer only.