  The choice of a homogeneous ideal in a polynomial ring defines a closed
subscheme $Z$ in a projective space as well as an infinite sequence of cones
over $Z$ in progressively higher dimension projective spaces. Recent work of
Aluffi introduces the Segre zeta function, a rational power series with integer
coefficients which captures the relationship between the Segre class of $Z$ and
those of its cones. The goal of this note is to define a relative version of
this construction for closed subschemes of projective bundles over a smooth
variety. If $Z$ is a closed subscheme of such a projective bundle $P(E)$, this
relative Segre zeta function will be a rational power series which describes
the Segre class of the cone over $Z$ in every projective bundle "dominating"
$P(E)$. When the base variety is a point we recover the absolute Segre zeta
function for projective spaces. Part of our construction requires $Z$ to be the
zero scheme of a section of a bundle on $P(E)$ of rank smaller than that of $E$
that is able to extend to larger projective bundles. The question of what
bundles may extend in this sense seems independently interesting and we discuss
some related results, showing that at a minimum one can always count on direct
sums of line bundles to extend. Furthermore, the relative Segre zeta function
depends only on the Segre class of $Z$ and the total Chern class of the bundle
defining $Z$, and the basic forms of the numerator and denominator can be
described. As an application of our work we derive a Segre zeta function for
products of projective spaces and prove its key properties.
