Torsten Ekedahl: The Sato-Tate conjecture

Time: Wed 2010-11-03 16.00

Location: Room 14, house 5, Kräftriket, Department of Mathematics, Stockholm University

The Sato-Tate conjecture is an equidistribution conjecture for certain
number-theoretically defined sequences. An example of a (generalised)
Sato-Tate conjecture is obtained by defining
$$
\sum_{n=1}^\infty \tau(n) = q\prod_{i=1}^\infty(1-q^n)^{24}.
$$
The conjecture then says that $\{\tau(n)/(2n^{5.5})\}$ is
equi-distributed with respect to a specific well-known distribution.
This is as well as the original Sato-Tate distribution has now been
proved by the combined efforts by a fairly large group of people.

I will mainly discuss how one by experimentation and
pseudo-probabilistic reasoning can arrive at the Sato-Tate conjecture
and then indicate the basic idea for the proof.