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In his influential paper on quantum modular forms, Zagier developed aconjectural framework describing the behavior of certain quantum knotinvariants under the action of the modular group on their argument. Moreprecisely, when $J_{K,0}$ denotes the colored Jones polynomial of a knot $K$,Zagier's modularity conjecture describes the asymptotics of the quotient$J_{K,0} (e^{2 \pi i \gamma(x)}) / J_{K,0}(e^{2 \pi i x})$ as $x \to \infty$along rationals with bounded denominators, where $\gamma \in\mathrm{SL}(2,\mathbb{Z})$. This problem is most accessible for thefigure-eight knot $4_1$, when the colored Jones polynomial has an explicitexpression in terms of the $q$-Pochhammer symbol. Zagier also conjectured thatthe function $h(x) = \log (J_{4_1,0} (e^{2 \pi i x}) / J_{4_1,0}(e^{2 \pi i/x}))$ can be extended to a function on $\mathbb{R}$ which is continuous atirrationals. In the present paper, we prove Zagier's continuity conjecture forall irrationals which have unbounded partial quotients in their continuedfraction expansion. In particular, the continuity conjecture holds almosteverywhere on the real line. We also establish a smooth approximation of $h$,uniform over all rationals, in accordance with the modularity conjecture. As anapplication, we find the limit distribution (after a suitable centering andrescaling) of $\log (J_{4_1,0}(e^{2 \pi i x}))$, when $x$ ranges over allreduced rationals in $(0,1)$ with denominator at most $N$, as $N \to \infty$,thereby confirming a conjecture of Bettin and Drappeau.

A conjecture of Zagier and the value distribution of quantum modular forms