Proving theorems and special cases (Part 9): The Riemann hypothesis

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve already seen that a conjecture could be true for the first 40 cases or even the first yet ultimately prove false for all cases.

For the next few posts, I thought I’d share a few of the most famous unsolved problems in mathematics… and just how much computational work has been done to check for a counterexample.

4. The Riemann hypothesis (see here, here, and here for more information) is perhaps the outstanding unsolved problem in pure mathematics, and a prize of $1 million has been offered for its proof.

The Riemann zeta function is defined by

for complex numbers with real part greater than 1. For example,

and

The definition of the Riemann zeta function can be extended to all complex numbers (except a pole at ) by the integral

and also analytic continuation.

The Riemann hypothesis conjectures to all solutions of the equation other than negative even integers occur on the line . At present, it is known that the first 10 trillion solutions are on this line, so that every solution with is on this line. Of course, that’s not a proof that all solutions are on this line.

A full description of known results concerning the Riemann hypothesis requires much more than a single post. I’ll refer the interested reader to the links above from MathWorld, Wikipedia, and Claymath and the references embedded in those links. An excellent book for the layman concerning the Riemann hypothesis is Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, by John Derbyshire.