In this note we study SL(2,Z)-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory. The functions solve inhomogeneous Laplace equations and we choose to represent them as Poincaré series. In this way we can combine different methods for asymptotic expansions and obtain the perturbative and non-perturbative contributions to their zero Fourier modes. In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/anti-instantons.
Dorigoni, D., & Kleinschmidt, A. (2019). Modular graph functions and asymptotic expansions of Poincare' series. Communications in Number Theory and Physics, 13(3), 569-617