  We present high-accuracy calculations of the density of states using
multicanonical methods for lattice gauge theory with a compact gauge group U(1)
on 4^4, 6^4 and 8^4 lattices. We show that the results are consistent with weak
and strong coupling expansions. We present methods based on Chebyshev
interpolations and Cauchy theorem to find the (Fisher's) zeros of the partition
function in the complex beta=1/g^2 plane. The results are consistent with
reweighting methods whenever the latter are accurate. We discuss the volume
dependence of the imaginary part of the Fisher's zeros, the width and depth of
the plaquette distribution at the value of beta where the two peaks have equal
height. We discuss strategies to discriminate between first and second order
transitions and explore them with data at larger volume but lower statistics.
Higher statistics and even larger lattices are necessary to draw strong
conclusions regarding the order of the transition.
