### 21 september 2006,
John Friedlander (University of Toronto, Canada),
Hyperbolic Prime Number Theorem

**Abstract.**
It is known since Fermat and Euler that prime numbers of
the form $4n+1$ are precisely the ones (in addition to 2) which can
be written as the sum of two squares. Because of the simplest case of the
prime ideal theorem this means that we can count asymptotically the
number of primes $p=x^2+y^2$ within a large disc $x^2+y^2 \le X$ in the
Euclidean plane.
In joint work with Henryk Iwaniec we study some natural generalizations
of this question with particular emphasis on analogues concerning points in
a large disc in the hyperbplic plane.