Geometric Prime Number Theorems
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/logT 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 with many illustrations.