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

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erdosproblems.com forum thread for Problem 1049 evidence for this event

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

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

This event attributed to Costa Lambrinoudis

Formal Conjectures: Erdős 1049 (open; integer-base variant) evidence for this event

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Erdos Problem 1049 evidence for this event

Twitter Announcement evidence for this event

W. Cook, Geometric Moments and Rational Lambert Values (earlier 0.405683 threshold) evidence for this event

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 $\lambda=(42963+428220/\pi^2)/202140\approx0.42718$. parent of this event

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