Resurgence with quasi-modularity: outperforming Ramanujan

I describe recent progress on two intriguing problems. The first concerns improvements on Ramuajan's evaluation of odd zeta values, by expansion in $\exp (-2 \pi) < 1/535$. The second concerns a study of resurgence in the asymptotic expansion of Lambert series, where easily computable perturbative terms are accompanied by non-perturbative corrections in $\exp (-1/x)$ at small $x$. I shall explain how the pioneering work of Spencer Bloch, Pierre Vanhove and Matt Kerr, on elliptic polylogarithms from Feynman integrals, led to progress on both problems, by exploiting quasi-modularity.