We develop the relation between hyperbolic geometry and arithmetic
equidistribution problems that arises from the action of arithmetic groups on
real hyperbolic spaces, especially in dimension up to 5. We prove
generalisations of Mertens' formula for quadratic imaginary number fields and
definite quaternion algebras over the rational numbers, counting results of
quadratic irrationals with respect to two different natural complexities, and
counting results of representations of (algebraic) integers by binary
quadratic, Hermitian and Hamiltonian forms with error bounds. For each such
statement, we prove an equidistribution result of the corresponding
arithmetically defined points. Furthermore, we study the asymptotic properties
of crossratios of such points, and expand Pollicott's recent results on the
Schottky-Klein prime functions.Comment: 44 page
