  The paper investigates vanishing conditions on the first cohomology module of
a normalized rank 2 vector bundle E on P^3 which force E to split, and finds
therefore strategic levels of non-vanishing for a non-split bundle. The present
conditions improve other conditions known in the literature and are obtained
with simple computations on the Euler characteristic function, avoiding the
speciality lemma, Barth's restriction theorem, the discriminat property, and
other heavy tools.
