Hermite–Padé approximation and the number $\pi$

It is well known that $\pi$ is an irrational number and that it is also a transcendental number, but there are still many open problems for this interesting number. It is still not known how well it can be approximated by rational numbers. Last year Zeilberger and Zudilin (2020) found the best upper bound so far: the measure of irrationality of $\pi$ is bounded from above by $7.103205334137$. They improved an earlier upper bound of Salikhov from 2008, and before him the best upper bound was obtained by Hata (1993). These upper bounds were obtained by analyzing certain integrals of rational functions over contours in the complex plane.

In my talk I will show that these integrals are closely related to an Hermite–Padé approximation problem for a pair of Markov functions. We will investigate this Hermite–Padé approximation in some detail using the corresponding vector equilibrium problem, algebraic functions satisfying a cubic equation and we describe the Riemann–Hilbert problem for this Hermite–Padé approximation problem.