On congruence conjectures of Andrews and Bachraoui

Abstract: Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle these conjectures by relating the corresponding generating function to modular forms and mock theta functions.

• The paper proves new congruences for partition coefficients using modular and mock modular form techniques.
• It employs q-series dissections and explicit generating function decompositions to establish rigorous divisibility properties.
• The study extends known conjectures and reveals infinite families of congruences, enhancing our understanding of partition arithmetic.

Analysis of "On congruence conjectures of Andrews and Bachraoui" (2604.02239)

Overview

This paper provides a rigorous resolution of several congruence conjectures concerning partition functions associated with restricted two-color partitions, as originally posited by Andrews and Bachraoui. The authors establish explicit congruences for the coefficients of certain $q$-series, which are linked deeply to mock theta functions, modular forms, and their dissections. They further derive structural identities for relevant generating functions, analyze their connection to both classically modular and mock modular forms, and extend the congruence framework with new infinite families.

Main Results

The central object of study is the generating function $C(q)$ for restricted two-color partitions with specific difference and congruence conditions, as formalized in Andrews and Bachraoui's earlier work. Explicitly, $C(q)$ and its variants are expressed through infinite $q$-products and specialized $q$-hypergeometric sums, and are deeply related to mock theta functions, in particular, Ramanujan's third order function $\omega$ and McIntosh's second order function $B$.

The paper resolves the following conjectures for the partition function coefficients $c(n)$ derived from $C(q)$:

• $c(8n+4) \equiv 0 \pmod{4}$ for all $C(q)$0,
• $C(q)$1 for all $C(q)$2,

Further, the authors prove a new congruence:

• $C(q)$3 for all $C(q)$4.

For the sequence $C(q)$5, arising as coefficients of a secondary generating function $C(q)$6 related to $C(q)$7, the previously conjectured vanishing property $C(q)$8 for all $C(q)$9 is also proven.

These results are achieved by relating $C(q)$0 explicitly to linear combinations of known mock theta functions and eta quotients, establishing their modular or mock modular behavior as anticipated by Andrews and Bachraoui.

Technical Approach

• Connections to Mock Theta Functions: The key insight is relating the generating functions for restricted partitions to mock theta functions. It is demonstrated that $C(q)$1 can be written in terms of the second-order mock theta function $C(q)$2 and eta quotients:

$C(q)$3

This identity forms the backbone for modular dissection and modular form techniques applied in the proof of congruences.

• Dissection and Modular Techniques: By employing $C(q)$4-series dissections modulo small powers of $C(q)$5, the authors analyze the parity and congruence classes of coefficients in $C(q)$6 and $C(q)$7. They utilize classical identities for theta functions, such as Ramanujan's and Watson's relations, as well as modular transformations and properties of harmonic Maass forms.
• Explicit Construction and Algebraic Sieving: The proofs systematically decompose generating functions into combinations of modular forms whose $C(q)$8-expansion coefficients satisfy strong divisibility properties, enabling the extraction of the desired congruences.
• Vanishing and New Congruences: The proof of the vanishing property for $C(q)$9 is accomplished via a detailed exploitation of modular and mock modular identities, and by leveraging explicit expressions for the relevant generating series.

Numerical Strength and Claims

The congruences proved in the paper—such as $q$0 and $q$1—mirror Ramanujan-type partition congruences and exhibit strong regularity modulo small primes. The approach is constructive and algebraic rather than solely computational, so the congruence proofs are valid for all $q$2.

The authors present an extended conjectural framework, predicting new infinite families of congruences for $q$3 modulo $q$4 and $q$5, e.g.,

$q$6

and similar statements for higher moduli, checked numerically up to $q$7 so that the indices are bounded above by $q$8.

Implications and Future Directions

These results significantly reinforce the understanding of arithmetic properties of partition functions defined via combinatorial constraints connected to mock modular and modular forms. The explicit connection to mock theta functions and eta quotients strengthens the methodology for deducing divisibility and vanishing properties of such sequences, situating the explored generating functions within the broader framework of harmonic Maass forms.

The infinite families of congruence conjectures suggest a deeper layer of modularity or arithmetic symmetry, potentially pointing to hidden Hecke algebra actions, or the existence of natural lifts to higher rank $q$9-series or automorphic forms.

Future work may explore:

• Systematic classification of all possible congruences for these partition functions at higher moduli,
• Analytical interpretations of these congruences in paramodular or Siegel modular settings,
• Algorithmic or computational techniques for proving conjectured infinite family congruences,
• Connections to physical models, e.g., in statistical mechanics, where such constrained partition functions occur.

Conclusion

This paper provides definitive proofs of congruence and vanishing conjectures for coefficients of the generating functions associated with restricted two-color partitions. By expressing the generating functions in terms of mock theta functions and eta quotients, the authors unlock structural identities yielding nontrivial and highly regular congruence relations. The work advances both the theory of mock modular forms and the arithmetic of partition functions, and opens up new avenues for exploring congruences in similar combinatorially-defined sequences.