The prime number theorem states that the number of primes of size at most T grows like T/log T, proved by Hadamard and de la Vallee Poussin in 1896. For Gaussian primes, that is, prime ideals in Z[i], not only does the number of Gaussian primes of norm at most T grow like T/log T but also the angular components of Gaussian primes are equidistributed in all directions, as proved by Hecke in 1920.

Geometric analogues of these profound facts have been of great interest over the years. We will discuss effective versions of these theorems for hyperbolic 3-manifolds and for rational maps. (The class of quadratic polynomials to which our theorem applies forms an open dense subset of the Mandelbroat set conjectually.)

Both Kleinian groups and rational maps define dynamical systems on the Riemann sphere and they are expected to behave analogously in view of Sullivan’s dictionary. We will also explain how our theorems fit in this dictionary. This talk is based on joint work with Winter, and in part with Margulis and Mohammadi.