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Claim. K. Postelmans and W. Van Assche, Irrationality of ζ_q(1) and ζ_q(2), J. Number Theory 126 (2007), no. 1, 119--154 (arXiv:math/0604312, 13 April 2006, the date this page carries). Theorem 1.3 (p. 3 of the arXiv version): for $q=1/p$ with $p\in\{2,3,\dots\}$, the numbers $1$, $\zeta_q(1)$ and $\zeta_q(2)$ are linearly independent over $\mathbb Q$, where $\zeta_q(s)=\sum_{n\ge1}n^{s-1}q^n/(1-q^n)$; the paper's Theorem 1.2 with its Lemma 1.1 already gives the irrationality of $\zeta_q(2)$. At $p=2$, $\zeta_{1/2}(2)=\sum_{n\ge1}\sigma(n)/2^n$, the series of Problem 250, so the theorem answers the question yes by a proof independent of Duverney's and Nesterenko's, whose results the paper's introduction cites.

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

Acceptance. Refereed: Journal of Number Theory, volume 126. The proof is recorded by statement and pointer only.