  A linear series on a curve C in $P^3$ is "primary" when it does not contain
the series cut by planes. We provide a lower bound for the degree of these
series, in terms of deg(C), g(C) and of the number $s = min{i: h^0(I_C(i))\neq
0}$; as a corollary, we obtain bounds for the gonality of C. Examples show that
our bound is sharp. Extensions to the case of general linear series and to the
case of curves in higher projective spaces are considered.
