  The paper is concerned with the Kontsevich-Zagier formal power series $$
f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n) $$ and its analytic properties. To
begin with, we give an explicit formula for the Borel transform of the
associated formal power series $F(x)=e^{-1/(24x)}f(e^{-1/x})$ from which its
analytic continuation, its singularities and their structure can be manifestly
determined. This gives rise to right/left and median summation of the original
power series. These sums, which are well-defined in the open right half-plane
are expressed by an integral formula involving the Dedekind eta function. The
median sum can also be expressed as a series involving the complex error
function. Moreover, it is shown using results of Zagier that the limiting
values at $-1/(2 \pi i \a)$ for rational numbers $\a$ coincide with $F(-1/(2
\pi i \a))$. One motivation for studying the series $f(q)$ is Quantum Topology,
which assigns numerical invariants to knotted 3-dimensional objects. Motivated
by our results, we formulate a resurgence conjecture for the formal power
series of knotted objects, which we prove in the case of the trefoil knot and
the Poincare homology sphere, and more generally for torus knots and Seifert
fibered 3-manifolds. In a subsequent publication we will study resurgence for a
class of power series that includes the quantum invariants of the simplest
hyperbolic $4_1$ knot.
