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Claim. Write $F(t)=\sum_{n\ge1}(t^n-1)^{-1}$. For coprime integers $a>b\ge1$ with

$$\frac{\log b}{\log a}<\frac12-\frac1{\pi^2}=0.39868\ldots,$$

$F(a/b)$ is irrational, with an irrationality measure. This is the case $K=\mathbb Q$, $v=\infty$, $\alpha=-1$ of Theorem 2 of P. Bundschuh and K. Väänänen, Arithmetical investigations of a certain infinite product, Compositio Math. 91 (1994), no. 2, 175--199 (p. 177). The paper sets $E_q(z)=\prod_{j\ge1}(1+zq^{-j})$ and, by logarithmic differentiation, $L_q(z)=E_q'(z)/E_q(z)=\sum_{j\ge1}(q^j+z)^{-1}$, so $L_q(-1)=F(q)$, and the irrationality of $L_q(\alpha)$ is equivalent to the linear independence of $E_q(\alpha)$ and $E_q'(\alpha)$ over $\mathbb Q$ (p. 177). Theorem 2 bounds $|a_0E_q(\alpha)+a_1E_q'(\alpha)|$ from below by an explicit power of the height of $(a_0,a_1)$ when $\lambda=\log h(q)/\log|q|$ is small enough, and in the case $\alpha=-1$ it allows every $\lambda<(1/2+1/\pi^2)^{-1}$. For $q=a/b$ in lowest terms $h(q)=a$, so $\lambda=\log a/\log(a/b)$, and the condition on $\lambda$ is the condition on $\log b/\log a$ above. The proof (Section 3) works from linear forms $J(n)=P_0E_q(\alpha)-\alpha P_1E_q'(\alpha)$ with polynomial coefficients, whose size it controls under the same condition, and the paper reads its bounds as irrationality measures for $L_q(\alpha)$ (p. 178). The year of the volume is the only date the record gives, so this page is dated to its first day.

Covers. Every $t=a/b>1$ in lowest terms with $\log b/\log a<\frac12-\frac1{\pi^2}$ of Problem 1049: every integer $t\ge2$ ($b=1$), already settled by Erdős, and infinitely many non-integer rationals, for example $7/2$, since $\log2/\log7=0.356\ldots$. It does not cover $3/2$, where $\log2/\log3=0.63\ldots$.

Acceptance. Refereed: the paper is a journal publication in Compositio Mathematica, volume 91, no. 2 (1994), received 6 October 1992, the refereed evidence. The site's page does not cite it; the forum comment of 11 September 2026 (the discussion link) points to its Theorem 2 for this region. The site labels the problem OPEN, so no reviewed evidence is listed.

Depends on. Nothing in this wiki.