RESURGENCE OF THE KONTSEVICH-ZAGIER SERIES

The paper is concerned with the resurgence of the Kontsevich-Zagier series f(q) = Sigma(infinity)(n=0)(1 - q) ... (1 - q(n)) We give an explicit formula for the Borel transform of the power series when q = e(1/x) from which its analytic continuation, its singularities (all on the positive real axis) and the local monodromy can be manifestly determined. We also give two formulas (one involving the Dedekind eta function, and another involving the complex error function) for the right, left and median summation of the Borel transform. We also prove that the limiting values of the median sum at rational multiples of 1/(2 pi i) coincide with the values of f(q) at the corresponding complex roots of unity. Our resurgence theorem extends more generally to the power series of torus knots and Seifert fibered 3-manifolds associated by Quantum Topology.