Communications in Number Theory and Physics

Volume 15 (2021)

Number 1

Resurgent expansion of Lambert series and iterated Eisenstein integrals

Pages: 1 – 57



Daniele Dorigoni (Centre for Particle Theory & Department of Mathematical Sciences, Durham University, Durham, United Kingdom)

Axel Kleinschmidt (Max-Planck-Institut für Gravitationsphysik, Potsdam, Germany; and International Solvay Institutes, Brussels, Belgium)


We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity properties of iterated Eisenstein integrals that have recently attracted attention in the context of certain period integrals and string theory scattering amplitudes.


Lambert series, Eisenstein integrals, string scattering amplitudes, asymptotic expansion

Received 5 February 2020

Accepted 5 June 2020

Published 4 January 2021