Sign Laws and Mock Theta Functions

Technical Summary: Sign Laws and Mock Theta Functions

Problem Statement
This paper addresses the sign behavior of the coefficients $r(n)$ of Ramanujan's third-order mock theta function $\rho(q)$, defined by the series:
$$\rho(q) = \sum_{m \ge 0} \frac{q^{2m(m+1)}}{(1 + q + q^2)(1 + q^3 + q^6) \cdots (1 + q^{2m+1} + q^{4m+2})} = \sum_{n \ge 0} r(n)q^n.$$
Empirical observation of the initial coefficients reveals a striking periodic sign pattern modulo 3: $r(3m) > 0$, while $r(3m+1) \le 0$ and $r(3m+2) \le 0$. The zeros in the non-positive residue classes appear to occur only at specific small integers ($n = 2, 4, 8, 11, 20$). The primary objective is to rigorously prove this "sign law" for all $n \ge 0$ and to identify the exact set of indices where $r(n) = 0$.

Methodology
The proof combines analytic number theory (specifically the circle method adapted for mock theta functions) with exact integer arithmetic.

1. Reduction via Watson's Identity:
   The authors utilize Watson's identity relating $\rho(q)$ to another third-order mock theta function $\omega(q)$ and an eta-quotient $T(q)$:
   $$2\rho(q) + \omega(q) = T(q), \quad \text{where } T(q) = 3 \frac{(q^6; q^6)_\infty^4}{(q^3; q^3)_\infty^2 (q^2; q^2)_\infty}.$$
   This reduces the problem to analyzing the difference $T(q) - \omega(q)$. The coefficients of $T(q)$ are modular, while $\omega(q)$ is a mock theta function.

2. Circle Method and Root-of-Unity Analysis:
   The coefficients $r(n)$ are extracted via Cauchy's integral formula on a circle $|q| = e^{-\beta}$. The integral is decomposed into "major arcs" near roots of unity and "minor arcs."

   • Cancellation at $q=1$: The leading polar terms of $T(q)$ and $\omega(q)$ at $q=1$ are identical and cancel out in the difference, leaving a polynomially bounded contribution.
   • Vanishing at $q=-1$: The contribution at $q=-1$ is also shown to be polynomially bounded.
   • Dominance at Cubic Roots: The first surviving exponential terms arise from the primitive cubic roots of unity ($\zeta = e^{2\pi i/3}$ and $\zeta^2$). Using the Zwegers completion and transformation laws, the authors derive effective asymptotic estimates for $\rho(q)$ near these roots.

3. Effective Asymptotics:
   The contribution from the cubic arcs yields a main term of the form:
   $$\kappa_{n \pmod 3} \frac{2\pi}{d_n^{1/4}} I_{1/2}(x),$$
   where $d_n = 12n+8$, $x = \pi\sqrt{d_n}/18$, and the constants $\kappa_0, \kappa_1, \kappa_2$ are derived from trigonometric values involving $\pi/18$. Specifically:

   • $\kappa_0 = \frac{1}{3}\cos(\frac{\pi}{18}) > 0$
   • $\kappa_1 = -\frac{1}{3}\sin(\frac{2\pi}{9}) < 0$
   • $\kappa_2 = -\frac{1}{3}\sin(\frac{\pi}{9}) < 0$

   The authors establish explicit error bounds for the minor arcs and the non-dominant parts of the major arcs. They prove that for sufficiently large $n$, the main term dominates the error term, thereby determining the sign of $r(n)$.

4. Finite Verification:
   To handle the range of $n$ where the asymptotic dominance has not yet been proven, the authors perform an exact computation using integer arithmetic. They implement an algorithm in the truncated polynomial ring $\mathbb{Z}[q]/(q^{L+1})$ to compute the coefficients $r(n)$ exactly up to the crossover point determined by the analytic estimates.

Key Results
The paper proves Theorem 1.1, which establishes the sign law for all $n \ge 0$:

• $r(3m) > 0$ for all $m \ge 0$.
• $r(3m+1) \le 0$ and $r(3m+2) \le 0$ for all $m \ge 0$.
• Equality ($r(n)=0$) occurs precisely at $n \in \{2, 4, 8, 11, 20\}$.

The analytic part of the proof guarantees the sign law for $n \ge 392,275$. The exact integer verification confirms the result for $0 \le n < 392,275$, identifying the specific zeros and confirming no sign violations exist in this finite range.

Significance and Claims
The paper claims that the sign behavior of mock theta coefficients can be controlled by a "root-of-unity dominance principle." In the case of $\rho(q)$, the cancellation of the dominant pole at $q=1$ shifts the determining factor to the cubic roots of unity, whose phases dictate the periodic sign pattern.

The authors suggest that this methodology is not isolated to $\rho(q)$. They indicate that the same strategy—identifying the first non-cancelling root-of-unity contribution and controlling the remaining arcs—should apply to other third-order mock theta functions. Specifically, they cite preliminary evidence for analogous sign laws in Ramanujan's functions $\phi(q)$ (period 4) and $\chi(q)$ (period 6), proposing a systematic program to extend these results to all of Ramanujan's mock theta functions. The work bridges the gap between the qualitative modular properties of mock theta functions and precise quantitative arithmetic properties of their coefficients.