  The paper investigates the non-vanishing of $H^1(E(n))$, where $E$ is a
(normalized) rank two vector bundle over any smooth irreducible threefold $X$
of degree $d$ such that $Pic(X) \cong \ZZ$. If $\epsilon$ is the integer
defined by the equality $\omega_X = O_X(\epsilon)$, and $\alpha$ is the least
integer $t$ such that $H^0(E(t)) \ne 0$, then, for a non-stable $E$ ($\alpha
\le 0$) the first cohomology module does not vanish at least between the
endpoints $\frac{\epsilon-c_1}{2}$ and $-\alpha-c_1-1$. The paper also shows
that there are other non-vanishing intervals, whose endpoints depend on
$\alpha$ and also on the second Chern class $c_2$ of $E$. If $E$ is stable the
first cohomology module does not vanish at least between the endpoints
$\frac{\epsilon-c_1}{2}$ and $\alpha-2$. The paper considers also the case of a
threefold $X$ with $Pic(X) \ne \ZZ$ but $Num(X) \cong \ZZ$ and gives similar
non-vanishing results.
