Nonquadraticity of Lambert series values and improved bounds for Erdős Problem 1049

For rational bases $t=a/b>1$, $F(t)=\sum_{n\ge1}(t^n-1)^{-1}$ is irrational whenever $\log b/\log a<0.5728$, and a formalized refinement reaches $0.573290$. The previous thresholds were $1/2-1/\pi^2\approx0.3987$ (Bundschuh and Väänänen, 1994) and $0.405683$ (W. Cook, September 2026, self-published). Problem 1049 itself stays open: $t=3/2$, with $\log2/\log3\approx0.631$, is still outside the region. Along the way, for every integer $q\ge2$ the numbers $1,F(q),F(q)^2$ are linearly independent over $\mathbb Q$, so the Erdős-Borwein constant $F(2)$ is not quadratic irrational. Both follow from one degree criterion for real algebraic bases: no relation of degree at most $e$ over $\mathbb Q(q)$ when $e\lambda\log M(q)<\log q$, with $\lambda=(42963+428220/\pi^2)/202140\approx0.42718$.

For rational bases $t=a/b>1$, $F(t)=\sum_{n\ge1}(t^n-1)^{-1}$ is irrational whenever $\log b/\log a<0.5728$, and a formalized refinement reaches $0.573290$. The previous thresholds were $1/2-1/\pi^2\approx0.3987$ (Bundschuh and Väänänen, 1994) and $0.405683$ (W. Cook, September 2026, self-published). Problem 1049 itself stays open: $t=3/2$, with $\log2/\log3\approx0.631$, is still outside the region. Along the way, for every integer $q\ge2$ the numbers $1,F(q),F(q)^2$ are linearly independent over $\mathbb Q$, so the Erdős-Borwein constant $F(2)$ is not quadratic irrational. Both follow from one degree criterion for real algebraic bases: no relation of degree at most $e$ over $\mathbb Q(q)$ when $e\lambda\log M(q)<\log q$, with $\lambda=(42963+428220/\pi^2)/202140\approx0.42718$.