On a conjecture of Andrews and almost alternating sign patterns

Technical Summary: On a Conjecture of Andrews and Almost Alternating Sign Patterns

Problem Statement
This paper addresses a sign phenomenon observed by George Andrews in specific $q$-series found in Ramanujan's Lost Notebook. While classical partition-theoretic $q$-series typically exhibit coefficients that grow monotonically or remain bounded, Andrews identified five series ($\sigma(q), v_1(q), v_2(q), v_3(q), v_4(q)$) with unusual behavior. Specifically, for $v_2(q), v_3(q),$ and $v_4(q)$, numerical evidence suggested that the coefficients exhibit "great sign regularity," appearing to alternate in sign with only a sparse set of exceptions. The paper aims to prove Andrews' conjectures regarding these three series and to establish a general framework explaining this phenomenon.

Methodology
The authors employ an adapted version of Wright's Circle Method, a technique well-suited for non-modular $q$-series. The methodology proceeds in three main stages:

1. Asymptotics near Roots of Unity: The authors analyze the generating functions $v_j(q)$ as $q$ approaches roots of unity. They utilize a generalized $q$-hypergeometric form $V_{r,b}(q)$ to encompass $v_2$ and $v_4$. By employing contour integration (specifically Watson's contour integral) and the saddle-point method, they derive precise asymptotic expansions for these functions as $q \to \zeta e^{-z}$ where $\zeta$ is a root of unity.

   • For roots of unity of order $m \not\equiv 2 \pmod 4$, the functions remain bounded.
   • For orders $m \equiv 2 \pmod 4$ (specifically the dominant pole at $-1$), the functions exhibit exponential growth coupled with oscillatory terms involving the Bloch–Wigner dilogarithm $D(e^{i\theta})$.

2. Coefficient Extraction: Using the derived asymptotics of the generating functions near the dominant pole, the authors apply the Circle Method to extract the asymptotic behavior of the coefficients $V_j(n)$. This involves decomposing the contour integral into a major arc (around the dominant pole) and minor arcs. The major arc contribution yields the dominant term, while the minor arcs are shown to contribute to a smaller error term.

3. Sign Pattern Analysis: The authors establish a general theorem (Theorem 6.4) demonstrating that any integer sequence satisfying a specific asymptotic form—characterized by an exponential growth factor, a parity-dependent sign factor $(-1)^n$, and an oscillatory cosine term—necessarily exhibits an "almost alternating sign pattern." They utilize discrepancy theory (specifically bounds on the distribution of sequences like $\sqrt{n} \pmod 1$) to prove that the zeros of the oscillatory term occur with density zero, ensuring the sign alternation holds for almost all $n$.

Key Contributions and Results

• Proof of Andrews' Conjectures: The paper provides rigorous proofs for the sign behavior of $v_2(q), v_3(q),$ and $v_4(q)$.

  • Theorem A: Confirms that the sequences $V_2(n), V_3(n),$ and $V_4(n)$ satisfy an almost alternating sign pattern. Specifically, $|V_j(n)| \to \infty$ as $n \to \infty$ outside a set of density zero, and for almost all $n$, $V_j(n)$ and $V_j(n+1)$ have opposite signs.
  • Theorem B: Provides precise asymptotic formulas for the coefficients. For $j \in \{2, 3, 4\}$, the coefficients behave as:
    $$V_j(n) \sim (-1)^n \alpha_j \frac{e^{2\sqrt{n}\text{Re}(\sqrt{W_j})}}{\sqrt{n}} \cos\left(2\sqrt{n}\text{Im}(\sqrt{W_j}) - \delta_j\right)$$
    where $W_j$ involves the Bloch–Wigner dilogarithm and $\alpha_j, \delta_j$ are explicit constants. This formula explicitly reveals the interplay between exponential growth and oscillatory behavior that governs the sign.

• Combinatorial Interpretations: The authors provide new combinatorial interpretations for the coefficients $V_2(n), V_3(n),$ and $V_4(n)$ in terms of partitions and overpartitions. Notably, they correct and refine the interpretation for $V_4(n)$, adding a condition regarding the largest part's congruence modulo 4, which was previously missing in Andrews' original formulation.

• Generalization: The results are extended to infinite families of $q$-hypergeometric series of the form $V_{r,b}(q)$ and linear combinations $U_{2,b}(q)$. The paper demonstrates that the sign phenomenon is not an isolated curiosity but arises systematically from the oscillatory asymptotics of these series near roots of unity.

• New Family and Numerical Observations: The authors introduce a new family of series $v_{\{k\}}(q)$ and present numerical evidence suggesting their coefficients exhibit a generalized sign pattern of length $k$. While not proven, this suggests the phenomenon is widespread.

Significance
The paper resolves long-standing conjectures by Andrews regarding the sign regularity of specific $q$-series from Ramanujan's Lost Notebook. By connecting these series to the arithmetic of quadratic fields (via the Bloch–Wigner dilogarithm) and automorphic objects, the work situates these examples within a broader theoretical framework connecting $q$-hypergeometric series, partitions, and modular forms. The development of a general framework for proving "almost alternating" sign patterns based on asymptotic oscillations offers a new tool for analyzing the coefficients of non-modular $q$-series. The authors note that while the results for $v_1(q)$ were previously established by Folsom, Males, Rolen, and Storzer, this paper addresses the remaining series and provides a unified approach for a broader class of functions.