On congruence conjectures of Andrews and Bachraoui

1. Problem Statement

The paper addresses open problems regarding the arithmetic properties of a specific sequence of coefficients, $c(n)$, derived from restricted two-color partitions.

• Context: Andrews and Bachraoui previously studied restricted two-color partitions, denoted $C(k, n)$, where the smallest part is odd and blue, and specific constraints apply to even parts. They defined a limiting generating function $C(q) = \lim_{k \to \infty} C_k(q) = \sum_{n \ge 0} c(n)q^n$.
• The Conjectures: In their prior work, Andrews and Bachraoui proposed several Ramanujan-type congruences for $c(n)$ and a vanishing identity for a related sequence $s(n)$ (coefficients of a function $S(q)$). Specifically, they conjectured:
  1. $c(8n + 4) \equiv 0 \pmod 4$
  2. $c(8n + 6) \equiv 0 \pmod 8$
  3. $c(16n + 13) \equiv 0 \pmod 4$ (a new conjecture by the authors of this paper)
  4. $s(4n + 1) = 0$ (a vanishing identity for the coefficients of $S(q)$)
• The Gap: While the generating functions were known, the modular nature of $S(q)$ and the proofs for these specific congruences remained unproven. The authors noted that $S(q)$ likely exhibits "mock-modular" behavior but lacked a rigorous connection to established mock theta functions.

2. Methodology

The authors employ a combination of $q$-series identities, modular form theory, and mock theta function theory to resolve the conjectures.

• Decomposition of Generating Functions:
  The core strategy involves expressing the generating function $C(q)$ in terms of known modular and mock-modular objects. The authors utilize the relation $C(q) = q A(q) S(q)$, where $A(q)$ is a modular form and $S(q)$ is the target function.
• Connection to Mock Theta Functions:
  The central technical breakthrough is proving Theorem 1.1, which establishes an exact identity relating $S(q)$ to Ramanujan's third-order mock theta function $\omega(q)$ and McIntosh's second-order mock theta function $B(q)$:
  $$S(q) = 2B(-q) - \frac{(q; q^2)_\infty^2}{(-q^2; q^2)_\infty} \omega(-q)$$
  This identity allows the authors to leverage known congruence properties of $B(q)$ and $\omega(q)$.
• Dissection and Modular Reduction:
  To prove the congruences, the authors perform $p$-dissections (specifically 4-dissections and 16-dissections) of the generating functions. They analyze the coefficients modulo 2, 4, and 8 by:
  1. Reducing infinite products modulo small integers (e.g., using $(q; q)_\infty \equiv (q^2; q^2)_\infty \pmod 2$).
  2. Utilizing known congruences for the coefficients of $\omega(q)$ and $B(q)$ (e.g., $c_\omega(8n+3) \equiv 0 \pmod 4$).
  3. Showing that certain terms in the dissection vanish or cancel out modulo the target integer.
• Operator Techniques:
  For the vanishing identity $s(4n+1)=0$, the authors define a linear operator $R$ that extracts coefficients of the form $4n+1$. They prove that the operator applied to the product of a theta function and $\omega(q)$ yields a specific modular form, allowing them to show the cancellation of terms in $S(q)$.

3. Key Contributions and Results

A. Proof of Conjectures 1.2 and 1.3 (Theorems 1.4 and 1.5)

The authors successfully prove the two main conjectures regarding $c(n)$:

• Result: $c(8n + 4) \equiv 0 \pmod 4$.
• Result: $c(8n + 6) \equiv 0 \pmod 8$.
• Mechanism: By substituting the identity from Theorem 1.1 into the expression for $C(q)$, they isolate the contributions of $B(-q)$ and $\omega(-q)$. They demonstrate that the coefficients of $A(q)B(-q)$ at indices $8n+3$ and $8n+5$ vanish modulo 2 and 4 respectively, while the $\omega$ term vanishes modulo 4 and 8 due to known properties of Ramanujan's mock theta function.

B. New Congruence (Theorem 1.6)

The authors discover and prove a new congruence:

• Result: $c(16n + 13) \equiv 0 \pmod 4$.
• Mechanism: This is proven by analyzing the 16-dissection of the product $A(q)B(-q)$ modulo 2, showing it reduces to a series with only exponents divisible by 16, and combining this with the known congruence $c_\omega(16n+12) \equiv 0 \pmod 4$.

C. Proof of the Vanishing Identity (Theorem 1.8)

The authors confirm the conjecture that $s(4n+1) = 0$.

• Result: The coefficient of $q^{4n+1}$ in $S(q)$ is identically zero.
• Mechanism: They define $M(q) = R(\Theta(q)\omega(q))$ and prove it is a specific modular form. By applying the operator $R$ to the decomposition of $S(q)$, they show that the contributions from the mock theta function $B(q)$ and the term involving $\omega(q)$ are exactly equal and opposite, resulting in a net zero.

D. Future Directions (Section 8)

The paper concludes by proposing a new infinite family of congruences for $c(n)$ modulo 4 and 8 (Conjecture 8.2), generalizing the results of Theorems 1.4–1.6.

4. Significance

• Resolution of Open Problems: The paper definitively settles the conjectures made by Andrews and Bachraoui, moving them from empirical observations to proven theorems.
• Bridge Between Partition Theory and Mock Modularity: The work provides a concrete example of how restricted partition functions, which often appear combinatorial in nature, are deeply connected to the theory of mock theta functions and harmonic Maass forms.
• Methodological Advancement: The paper demonstrates the power of combining classical $q$-series manipulations (dissections) with modern mock modular theory. It shows that even complex partition functions can be decomposed into simpler modular and mock-modular components to reveal arithmetic properties.
• Foundation for Future Research: By establishing the modular nature of $S(q)$ and proposing new infinite families of congruences, the paper sets the stage for further exploration into the arithmetic of restricted partitions and the structure of mock theta functions.

In summary, this paper transforms a set of numerical conjectures about partition coefficients into a rigorous theoretical framework using the tools of modular forms and mock theta functions, providing both proofs for existing conjectures and new insights into the arithmetic structure of these sequences.