Lambert series of the form $\sum_{n>0}a(n)q^n/(1-q^n)$
are ubiquitous in mathematical physics.
In particular, 2-loop sunrise and 3-loop banana Feynman diagrams yield
Lambert series with $a(n)$ of the form $\chi(n)/n^s$ where $\chi(n)$ is a Dirichlet character.
Resurgence concerns the singular limit as $|q|$ approaches 1. In the Feynman
cases we can control this limit, obtaining rapidly convergent expressions, since the
Lambert series are iterated integrals of holomorphic Eisenstein series twisted by
a character. We generalize this result, to include modular resurgent
structures found in topological-string observables.