  We construct new rational approximants of Euler's constant that improve those
of Aptekarev et al. (2007) and Rivoal (2009). The approximants are given in
terms of certain (mixed type) multiple orthogonal polynomials associated with
the exponential integral. The dual family of multiple orthogonal polynomials
leads to new rational approximants of the Gompertz constant that improve those
of Aptekarev et al. (2007). Our approach is motivated by the fact that we can
reformulate Rivoal's construction in terms of type I multiple Laguerre
polynomials of the first kind by making use of the underlying Riemann-Hilbert
problem. As a consequence, we can drastically simplify Rivoal's approach, which
allows us to study the Diophantine and asymptotic properties of the
approximants more easily.
