On the Double Lambert Series Conjecture of Andrews-Dixit--Schultz-Yee

This paper completes the proof of the Andrews-Dixit-Schultz-Yee conjecture regarding the parity of a double Lambert series, building upon preliminary ideas proposed by Amdeberhan, Andrews, and Ballantine in 2026.

The Gist

Imagine you are a detective trying to solve a very strange, invisible puzzle involving numbers that dance to the rhythm of a special beat called "q."

The Mystery: The "Odd" Function

In 2026, a team of famous math detectives (Andrews, Dixit, Schultz, and Yee) found a mysterious equation called Y(q). They suspected this equation had a secret personality trait: they thought it was an "odd function."

In the world of math, being "odd" is like a perfect mirror image that flips upside down. If you plug in a number, you get a result. If you plug in the negative of that number, you get the exact opposite result. It's a very symmetrical, balanced behavior.

The team wrote down their suspicion as Conjecture 1.1: "Y(q) is an odd function."

They had a map (a formula) to find the treasure, but they got stuck at the final turn. The map they had (Formula 1.1) was too twisty and confusing to prove the symmetry. They needed a new pair of glasses to see the path clearly.

The New Approach: Rearranging the Furniture

Enter Qianwen Fang, the author of this paper. Fang looked at the messy map and said, "Let's rearrange the furniture."

Instead of looking at the problem the old way, Fang reorganized the equation (Formula 1.2) into a different shape. It's like taking a jumbled pile of LEGO bricks and sorting them into neat, color-coded stacks. Suddenly, the structure became much clearer.

Fang didn't work alone in a vacuum. The paper mentions that in 2026, another team (Amdeberhan, Andrews, and Ballantine) had already started building the bridge toward the solution. Fang's job was to finish the last few planks of that bridge.

The Solution: The "Helper" Characters

To solve the puzzle, Fang introduced a cast of "helper" characters (auxiliary functions named Z, A, B, D1, and D2). Think of these as different tools in a toolbox:

1. The Breakdown: Fang took the big, scary equation Y(q) and broke it down into smaller, manageable pieces using these helpers.
2. The Swap: One of the key tricks was realizing that one helper, B1, was actually just a "mirror image" (using negative numbers) of another helper, A. It's like realizing that if you turn a left-handed glove inside out, it becomes a right-handed one. Once you know this, you can swap them around easily.
3. The Cancellation: By using a famous, well-known rule from the "Library of Number Magic" (a q-series identity), Fang showed that two of the big pieces (D1 and D2) were almost identical, except for a tiny, predictable difference.

The Grand Reveal

When Fang put all these pieces together, the messy equation Y(q) simplified beautifully. The complicated parts canceled each other out, leaving behind a result that perfectly matched the definition of an "odd function."

The Verdict: The mystery is solved! The equation Y(q) is indeed an odd function, just as the original team suspected.

A Side Note: The Twin Detective

The paper also mentions a fun twist: After Fang finished the work, they learned that two other mathematicians (Cui and Tang) had solved the exact same puzzle at the same time using a very similar method. It's like two detectives in different cities solving the same crime independently and arriving at the same conclusion at the same moment!

What's Next?

Fang ends the paper with a challenge for future detectives. They found a shortcut (Conjecture 3.1) that would make the proof even simpler, but it requires a "more elementary method" (a simpler, more direct way of thinking) to prove.

In a nutshell: This paper is the final chapter in a story where a complex mathematical guess was proven true by rearranging the numbers, using clever swaps, and applying classic rules of the universe to show that everything balances out perfectly.

Technical Summary

1. Problem Statement

The paper addresses Conjecture 1.1, proposed by Andrews, Dixit, Schultz, and Yee in their 2022 work [2]. The conjecture concerns the parity of a specific double Lambert series, $Y(q)$.

The series is defined as:
$$Y(q) := \sum_{m,n \ge 1} \frac{(-q)^{2mn+m}}{(1 + q^m)(1 - q^{2m-1})}$$
The conjecture asserts that $Y(q)$ is an odd function of $q$ for $|q| < 1$. In other words, $Y(-q) = -Y(q)$, implying that the power series expansion of $Y(q)$ contains only odd powers of $q$ (or equivalently, the coefficient of $q^n$ is zero for all even $n$).

Previous attempts to prove this using the representation derived in [1] (Equation 1.1) were incomplete due to difficulties in the final step of the derivation.

2. Methodology

Fang's approach builds upon the framework established by Amdeberhan, Andrews, and Ballantine (2026), who provided initial ideas but did not complete the proof. The methodology involves the following steps:

• Reformulation: Instead of using the problematic representation from [1], the author rewrites $Y(q)$ into a more tractable double sum form (Equation 1.2):
  $$Y(q) = \sum_{m \ge 1} \frac{(-1)^m q^m}{1 - q^{2m-1}} \sum_{k=2m}^{\infty} \frac{(-1)^k q^k}{1 - q^k}$$
• Decomposition into Auxiliary Functions: The proof relies on defining several auxiliary series to decompose $Y(q)$:
  • $Z(q)$: A truncated version of the inner sum.
  • $A(q)$ and $B(q)$: Double sums involving terms with denominators $(1+q^{2i+1})(1+q^{2j+1})$.
  • $B_1(q)$: A finite double sum variant of $B(q)$.
  • $D_1(q)$ and $D_2(q)$: Combinations of the original series and the auxiliary functions.
• Algebraic Manipulation: The author establishes the identity:
  $$Y(q) = D_1(q) - D_2(q) - A(q) + B_1(q)$$
  The goal is to show that the even parts of these functions cancel out or combine to form an odd function.
• Lemma Proofs:
  • Lemma 2.1: Proves the symmetry relation $B_1(q) = A(-q)$. This is achieved by expanding the geometric series in the denominators and re-indexing the summation variables to show that $B_1(q)$ transforms exactly into $A(-q)$.
  • Lemma 2.2: Establishes the identity $D_1(q) - D_2(q) = q(q^4; q^4)_\infty^4 / (q^2; q^2)_\infty^2 \times \sum \dots$. This step utilizes a well-known identity from $q$-series theory (specifically Entry 29 in standard references like Berndt's Ramanujan's Notebooks) regarding the summation $\sum \frac{x^n}{1-yq^n}$.

3. Key Contributions

• Completion of the Proof: The primary contribution is the rigorous completion of the proof for Conjecture 1.1, which had remained open since its proposal.
• New Representation: The paper successfully navigates the difficulties of the previous representation (1.1) by utilizing the alternative form (1.2) and the specific decomposition into $D_1, D_2, A, B_1$.
• Verification of Parity: By proving that $B_1(q) = A(-q)$ and analyzing the difference $D_1(q) - D_2(q)$, the author demonstrates that the resulting expression for $Y(q)$ satisfies the condition of being an odd function.
• Independent Confirmation: The paper notes that Cui and Tang independently proved the same conjecture using a similar approach shortly after the paper was written, validating the robustness of the method.

4. Results

The paper concludes that Conjecture 1.1 is true. Specifically:

1. The double Lambert series $Y(q)$ is indeed an odd function of $q$.
2. The identity $Y(q) = D_1(q) - D_2(q) - A(q) + B_1(q)$ holds, and the specific properties of these components ensure the cancellation of all even powers of $q$ in the final expansion.
3. The paper provides a concrete path for future research by proposing Conjecture 3.1, which suggests a simpler, more elementary identity: $Y(q) = D_2(q) - D_1(q)$. If proven, this would significantly simplify the current proof structure.

5. Significance

• Resolution of an Open Problem: This work resolves a specific conjecture in the theory of partitions and $q$-series, contributing to the understanding of the parity of coefficients in Lambert series.
• Methodological Advancement: The techniques used, particularly the decomposition of complex double sums into manageable auxiliary functions and the application of classical $q$-series identities, offer a template for tackling similar parity problems in combinatorial number theory.
• Contextual Relevance: The paper fits into the broader context of recent work by Andrews and collaborators on the combinatorial properties of partition functions and $q$-series, bridging the gap between conjecture and rigorous proof.

In summary, Qianwen Fang provides the missing logical steps to prove that the Andrews–Dixit–Schultz–Yee double Lambert series is an odd function, utilizing a clever decomposition strategy and standard $q$-series identities.