On a conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series

Abstract: In this note, we prove a recent conjecture of Amdeberhan, Andrews and Ballantine concerning a double Lambert series. More precisely, they conjectured that [ \coeff{q^{N2^a}} \sum_{m,k\geq 1} \frac{q^{mk2^a}}{(1+q^{k2^{a-1}})(1-q^{2m-1})} =σ_1(N), ] where $σ_1(N)$ is the sum of all the positive divisors of $N$. The main idea of the proof is to first transform a double Lambert series on the left-hand side into a single sum. This leads us to derive a new representation of quasi-modular forms $E_2(q)$.

• The paper proves the conjecture by reducing a double Lambert series to a single sum that exactly yields the divisor sum σ₁(N).
• It employs telescoping series and variable transformation to derive a new algebraic representation of the quasi-modular Eisenstein series E₂(q).
• The results provide new analytic tools for modular forms, suggesting potential generalizations to higher-weight Eisenstein series and related combinatorial identities.

Resolution of a Double Lambert Series Conjecture and New Quasi-Modular Representations

Introduction

The paper addresses a conjecture proposed by Amdeberhan, Andrews, and Ballantine concerning a double Lambert series, establishing a direct connection to divisor sums $\sigma_1(N)$—the sum of positive divisors of $N$. The resolution involves reducing a double Lambert series to a single sum, yielding a new algebraic representation of quasi-modular forms, specifically the Eisenstein series $E_2(q)$. Lambert series remain central to analytic number theory, $q$-series, and modular forms, and the generalization to double Lambert series continues to uncover deep arithmetic and combinatorial structures.

Double Lambert Series and Divisor Function Connection

The main conjecture proved is: $$[q^{N2^a}] \sum_{m,k\geq1} \frac{q^{mk2^a}}{(1+q^{k2^{a-1}})(1-q^{2m-1})} = \sigma_1(N)$$ This equates the coefficient of $q^{N2^a}$ in a specific double Lambert series to the sum of divisors of $N$. The proof involves clever manipulation of $q$-series identities, notably by transforming the double sum into a single Lambert series and demonstrating the equivalence via explicit coefficient extraction. The approach builds upon earlier works, including recent proofs for related double Lambert series conjectures (Cui et al., 10 Apr 2026, Fang, 5 Apr 2026).

Transformation Technique and $E_2(q)$ Representation

A critical step is the transformation: $$\sum_{m,k \geq 1} \frac{q^{2mk}}{(1+q^k)(1-q^{2m-1})} = \sum_{n\geq 1} \frac{q^{2n}}{(1-q^{2n})^2}$$ The right-hand side is the generating function for $N$0, from which the coefficient correspondence is derived. This transformation is executed via telescoping series and a change of variables, exploiting the algebraic structure of the $N$1-series. The result allows a new representation of the quasi-modular form: $N$2 where $N$3 is the (quasi-)modular Eisenstein series of weight 2.

This representation aligns with classical definitions but exposes the role of double Lambert series in quasi-modular form expansions and suggests potential for analogous constructions for higher-weight Eisenstein series.

Numerical Results and Contradictory Claims

The principal numerical result is the precise coefficient extraction, equating the double Lambert generating function to $N$4. The paper asserts the complete resolution of Conjecture 5.12 from [AAB], previously open for general $N$5, and notes the independent confirmation by other researchers. The transformation methodology is asserted to be new, especially the algebraic reduction of double sums and its application to modular form theory.

The corollary providing a new representation for $N$6 stands in contrast to classical expressions involving single-variable Lambert series, suggesting a broader algebraic framework for modular and quasi-modular forms.

Implications and Future Directions

The resolution of the conjecture enriches the understanding of Lambert series generalizations and their arithmetic properties. The transformation techniques introduced may be applicable for other double $N$7-series arising in combinatorial number theory and modular forms. Key questions proposed include:

• Whether a similar representation holds for generalized divisor functions $N$8 via double Lambert series, potentially providing alternative expansions for higher-weight Eisenstein series $N$9.
• The combinatorial and analytic consequences of such expansions, especially in the context of mock theta functions and partition identities.

A successful extension would yield new analytic tools for the study of modular forms, offering further insight into their arithmetic and structural properties.

Conclusion

The paper rigorously settles a previously conjectured identity connecting double Lambert series coefficients to divisor sums. The provided transformation enables a novel representation of $E_2(q)$0, illuminating new algebraic connections in $E_2(q)$1-series and modular forms. The methods presented suggest avenues for generalization and deeper exploration of double Lambert series and their role in analytic number theory and modular form theory.