Resurgent Lambert series from Feynman and beyond

Lambert series of the form ${\sum }_{n>0}a(n)q^n\mathrm{/}(1-q^n)$ 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 $\chi (n)\mathrm{/}{n}^s$ where $\chi (n)$ is a Dirichlet character. Resurgence concerns the singular limit as $\mathrm{\mid }q\mathrm{\mid }$ 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.