  Quantum fluctuations of the charge in the single electron box are
investigated. Based on a diagrammatic expansion we calculate the average island
charge number and the effective charging energy in third order in the tunneling
conductance. Near the degeneracy point where the energy of two charge states
coincides, the perturbative approach fails, and we explicitly resum the leading
logarithmic divergencies to all orders. The predictions for zero temperature
are compared with Monte Carlo data and with recent renormalization group
results. While good agreement between the third order result and numerical data
justifies the perturbative approach in most of the parameter regime relevant
experimentally, near the degeneracy point and at zero temperature the
resummation is shown to be insufficient to describe strong tunneling effects
quantitatively. We also determine the charge noise spectrum employing a
projection operator technique. Former perturbative and semiclassical results
are extended by the approach.
