Algebraic-degree and irrationality bounds for reciprocal Lambert series

For F(q)=Σₙ≥₁1/(qⁿ−1), determine algebraic degree and irrationality as q varies. The stated degree criterion excludes relations of degree at most e when eλ log M(q)<log q, with λ=(42963+428220/π²)/202140, giving integer-base nonquadraticity and explicit rational-base irrationality ranges.

Why it matters and what counts as progress

Why it matters: It asks for arithmetic information about a Lambert-series value beyond numerical approximation, including larger proven irrationality regions.

What counts as progress: Improve the excluded algebraic degree or rational-base irrationality range with rigorous degree criteria and certified numerical constants.

How the problem developed

1948 · Partial result [1]
Paul Erdős Proved irrationality for the reciprocal Lambert-series values at integer bases q greater than one.
1991 · Partial result [2]
Peter B. Borwein Extended irrationality results to reciprocal geometric sums with rational shifts, including the denominator qⁿ−1 case.
2026 · Reported claim [3]
Generative AI, directed and supervised by the repository author The repository reports linear independence of 1, F(q), and F(q)² over Q for integer q≥2, plus a degree-exclusion criterion for real algebraic bases.

AI contribution

Nonquadraticity of Lambert Series Values

Costa Lambrinoudis

For F(q)=Σₙ≥₁1/(qⁿ−1), a degree criterion excludes polynomial relations of degree at most e over Q(q) when eλ log M(q)<log q, where λ=(42963+428220/π²)/202140. It gives nonquadraticity for every integer q≥2 and irrationality for rational q=a/b>1 when log b/log a<0.5728; a certified refinement reaches 0.573290.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4.33.1 kernel with Mathlib, axiom audit and exact refinement certificates)

Human understandingFew