Sign Laws and Mock Theta Functions

Abstract

Let \[ \rho(q)=\sum_{m\geq 0}\frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n \] be Ramanujan's third order mock theta function. We prove the sign law \[ r(3m)>0,\qquad r(3m+1)\leq 0,\qquad r(3m+2)\leq 0, \] with equality precisely at $n=2,4,8,11,20$. Watson's identity \[ 2\rho(q)+\omega(q)=T(q) \] reduces the problem to comparing the mock theta function $\omega(q)$ with the eta quotient \[ T(q)=3\frac{(q^6;q^6)_\infty^4}{(q^3;q^3)_\infty^2(q^2;q^2)_\infty}. \] We prove effective root-of-unity estimates for this difference. The polar contributions at $q=1$ cancel, the contribution at $q=-1$ is polynomially bounded, and the first surviving exponential term occurs at the primitive cubic roots of unity. It has the sign pattern \[ \kappa_0=\frac13\cos\frac\pi{18}>0,\qquad \kappa_1=-\frac13\sin\frac{2\pi}{9}<0, \qquad \kappa_2=-\frac13\sin\frac\pi9<0. \] The resulting effective asymptotic proves the desired sign law for all sufficiently large $n$, and an exact integer-arithmetic verification completes the finite range. We conclude by indicating how the same root-of-unity method should lead to analogous sign laws for other third order mock theta functions, including $\phi(q)$ and $\chi(q)$.