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Let $\pi : X \to B$ be an elliptically fibered threefold satisfying $c_3(T_X\otimes \omega_X)=0$. We conjecture that the $\pi$-relative generating seriesof Pandharipande-Thomas invariants of $X$ are quasi-Jacobi forms and satisfytwo holomorphic anomaly equations. For elliptic Calabi-Yau threefolds ourconjectures specialize to the Huang-Katz-Klemm conjecture. The proposedformulas constitute the first case of holomorphic anomaly equations inPandharipande-Thomas theory.  We prove our conjectures for the equivariant Pandharipande-Thomas theory of$\mathbb{C}^2 \times E$ when specialized to the anti-diagonal action. For $K3\times \mathbb{C}$ we state reduced versions of our conjectures. As a corollarywe find an explicit conjectural formula for the stationary theory generalizingthe Katz-Klemm-Vafa formula for K3 surfaces. Further evidence is available for$\mathbb{P}^2 \times E$ based on earlier work of the second author.  To deal with elliptic threefolds with $c_3(T_X \otimes \omega_X) \neq 0$ weshow that the moduli space of $\pi$-stable pairs is represented by a properalgebraic space. We conjecture that the associated $\pi$-stable pair invariantsform quasi-Jacobi forms.

Pandharipande-Thomas theory of elliptic threefolds, quasi-Jacobi forms and holomorphic anomaly equations