The Shifted Lambert Coefficient

Abstract

Odd divisors encode the shifted Lambert coefficients.

Theorem 1.1 (Lambert coefficient and divisor count).

Lean statement: D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiLambert.lambert_divisor_coefficient

Proof. Machine-checked in Lean as D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiLambert.lambert_divisor_coefficient (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

Commentary.

For every natural number n, the coefficient of degree n in the finite shifted Lambert sum, increased by two, equals the number of positive divisors of 2n plus 5. The correspondence sends a summand indexed by j to the odd divisor 2j plus 3 and removes the two endpoint divisors.

References

• Truth anchor: D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiLambert.lambert_divisor_coefficient
• Dependency: D5/S3/Combinatorics/TwoColorPartition/AndrewsElBachraouiDefs