In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of given height.
If S is a set with height function H, such that there are only finitely many elements of bounded height, define a counting function and a zeta function If Z has abscissa of convergence β and there is a constant c such that N has rate of growth then a version of the Wiener–Ikehara theorem holds: Z has a t-fold pole at s = β with residue c.a.Γ(t).
The abscissa of convergence has similar formal properties to the Nevanlinna invariant and it is conjectured that they are essentially the same.
[1] Let X be a projective variety over a number field K with ample divisor D giving rise to an embedding and height function H, and let U denote a Zariski-open subset of X.
Then for every ε > 0 there is a U such that β < α + ε: in the opposite direction, if α > 0 then α = β for all sufficiently large fields K and sufficiently small U.