Resurgent Lambert series with characters

Abstract: We consider certain Lambert series as generating functions of divisor sums twisted by Dirichlet characters and compute their exact resurgent transseries expansion near $q=1^-$. For special values of the parameters, these Lambert series are expressible in terms of iterated integrals of holomorphic Eisenstein series twisted by the same characters and the transseries representation is a direct consequence of the action of Fricke involution on such twisted Eisenstein series. When the parameters of the Lambert series are generic the transseries representation provides for a quantum-modular version of Fricke involution which for a particular example we show being equivalent to modular resurgent structures found in topological strings observables.