  We discuss apparent paradoxes regarding the location of the zeros of the
partition function in the complex $\beta$ plane (Fisher's zeros) of a pure
SU(2) lattice gauge theory in 4 dimensions. We propose a new criterion to draw
the region of the complex $\beta$ plane where reweighting methods can be
trusted when the density of states is almost but not exactly Gaussian. We
propose new methods to infer the existence of zeros outside of this region. We
demonstrate the reliability of these proposals with quasi Gaussian Monte Carlo
distributions where the locations of the zeros can be calculated by independent
numerical methods. The results are presented in such way that the methods can
be applied for general lattice models. Applications to specific lattice models
will be discussed in a separate publication.
