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We obtain an exact modularity relation for the $q$-Pochhammer symbol. Usingthis formula, we show that Zagier's modularity conjecture for a knot $K$essentially reduces to the arithmeticity conjecture for $K$. In particular, weshow that Zagier's conjecture holds for hyperbolic knots $K\neq 7_2$ with atmost seven crossings.  For $K=4_1$, we also prove a complementary reciprocity formula which allowsus to prove a law of large numbers for the values of the colored Jonespolynomials at roots of unity. We conjecture a similar formula holds for allknots and we show that this is the case if one assumes a suitable version ofZagier's conjecture.

Modularity and value distribution of quantum invariants of hyperbolic knots