In mathematics, the height zeta function of an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

or more generally a subset of a variety encodes the distribution of points of given

height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...

Definition

If ''S'' is a set with height function ''H'', such that there are only finitely many elements of bounded height, define a ''counting function'' :

N(S,H,B) = \#\ .

and a ''zeta function'' :

Z(S,H;s) = \sum_ H(x)^ .

Properties

If ''Z'' has

abscissa of convergence In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of : \sum_^\infty a_n e^, where a_n, s are complex numbers and \ is a strictly increasing sequence of nonnegative real numbers that tends t ...

β and there is a constant ''c'' such that ''N'' has rate of growth :

N \sim c B^a (\log B)^

then a version of the

Wiener–Ikehara theorem The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (Chandrasekharan, 1969). Statement Let ''A''(''x'') be a non-negative, monotonic no ...

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 In mathematics, the Nevanlinna invariant of an ample divisor ''D'' on a normal projective variety ''X'' is a real number connected with the rate of growth of the number of rational points on the variety with respect to the embedding defined by the ...

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''.

References

* * {{cite book , first=Serge , last=Lang , authorlink=Serge Lang , title=Survey of Diophantine Geometry , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, year=1997 , isbn=3-540-61223-8 , zbl=0869.11051 Diophantine geometry Geometry of divisors