The Pauli operator describes the energy of a nonrelativistic quantum particle
with spin 1/2 in a magnetic field and an external potential. Bounds on the sum
of the negative eigenvalues are called magnetic Lieb-Thirring (MLT)
inequalities. The purpose of this paper is twofold. First, we prove a new MLT
inequality in a simple way. Second, we give a short summary of our recent proof
of a more refined MLT inequality \cite{ES-IV} and we explain the differences
between the two results and methods. The main feature of both estimates,
compared to earlier results, is that in the large field regime they grow with
the optimal (first) power of the strength of the magnetic field. As a byproduct
of the method, we also obtain optimal upper bounds on the pointwise density of
zero energy eigenfunctions of the Dirac operator.