Resurgent Lambert series with characters

TL;DR

This paper analyzes Lambert series twisted by Dirichlet characters, deriving their resurgent transseries expansions near q=1 and revealing connections to Eisenstein series, Fricke involution, and quantum modularity, with applications to topological string theory.

Contribution

It introduces a new resurgent transseries framework for Lambert series with characters, linking them to Eisenstein series and quantum modular forms.

Findings

Exact transseries expansions near q=1 for Lambert series with characters

Expression of special cases in terms of iterated Eisenstein integrals

Identification of quantum modular structures related to topological strings

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.