The inversion formula of polylogarithms and the Riemann-Hilbert problem

In this article, we set up a method of reconstructing to polylogarithms ${\mathrm{L}\mathrm{i}}_k(z)$ from zeta values $\zeta (k)$ via the Riemann-Hilbert problem. This is referred to as “a recursive Riemann-Hilbert problem of additive type.” Moreover, we suggest a framework of interpreting the connection problem of the Knizhnik-Zamolodochikov equation of one variable as a Riemann-Hilbert problem.