Height zeta function

In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of 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

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and a zeta function

$Z(S,H;s)=\sum _{x\in S}H(x)^{-s}.$

If Z has abscissa of convergence α and there is a constant c such that N has rate of growth

$N\sim cB^{a}(\log B)^{t-1}$

then a version of the Wiener–Ikehara theorem holds: Z has a t-fold pole at s = α 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.

References

• Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. Vol. 201. ISBN 0-387-98981-1. Zbl 0948.11023.