On linear independence of special values of Lerch functions

We shall discuss a recent joint work with Makoto Kawashima and Noriko Hirata-Kohno. For any set of algebraic numbers in a fixed number field $K$ satisfying standard metric conditions in the theory (close enough to zero), we prove that the values of Lerch functions with freely chosen shifts are linearly independent over $K$. This is done via a construction of an explicit system of Padé approximation. As a special case, we recover our previous results on polylogarithms.