Height zeta function

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. More precisely, Batyrev-Manin conjectured the following. 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. Let α = α(D) be the Nevanlinna invariant of D and β the abscissa of convergence of Z(U, H; s). 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.