  Given a multiarrangement of hyperplanes we define a series by sums of the
Hilbert series of the derivation modules of the multiarrangement. This series
turns out to be a polynomial. Using this polynomial we define the
characteristic polynomial of a multiarrangement which generalizes the
characteristic polynomial of an arragnement. The characteristic polynomial of
an arrangement is a combinatorial invariant, but this generalized
characteristic polynomial is not. However, when the multiarrangement is free,
we are able to prove the factorization theorem for the characteristic
polynomial. The main result is a formula that relates `global' data to `local'
data of a multiarrangement given by the coefficients of the respective
characteristic polynomials. This result gives a new necessary condition for a
multiarrangement to be free. Consequently it provides a simple method to show
that a given multiarrangement is not free.
