  A finite size scaling theory for the partition function zeros and
thermodynamic functions of O(N) phi^4-theory in four dimensions is derived from
renormalization group methods. The leading scaling behaviour is mean-field like
with multiplicative logarithmic corrections which are linked to the triviality
of the theory. These logarithmic corrections are independent of N for odd
thermodynamic quantities and associated zeros and are N dependent for the even
ones. Thus a numerical study of finite size scaling in the Ising model serves
as a non-perturbative test of triviality of phi^4_4-theories for all N.
