The Riemann hypothesis is perhaps the most famous unsolved problem in mathematics. It says that all non-trivial zeros of the Riemann zeta function have real part . What is the Riemann zeta function? A common misconception is that it is defined as
This is false! Because the series on the right only converges for . Also, it is a well-established fact that the trivial zeros of the Riemann zeta function occur at the negative even numbers . If we substitute in the above equation we get which is far from being zero. So even though these are called ‘trivial’ zeros, it seems highly non-trivial why these are actually zeros. What’s going on?
The problem lies in the above definition. This equation does NOT define the Riemann zeta function, which is a meromorphic function (see below) on the whole complex plane. The above definition only works for . Then what about the other values of ? Before proceeding further, let us familiarise ourselves with some terminology from the theory of complex analysis.
Complex analysis is one of the most beautiful branches of mathematics. It deals with functions in the complex plane that are differentiable in some open subset . (Open sets are like open intervals; open intervals are line segments excluding the endpoints, open sets in the complex plane can be thought of as discs excluding their boundaries.) Some basic definitions are:
- is said to be analytic (or holomorphic) at if it is differentiable in a neighbourhood of , i.e. at all points such that for some .
- above is called entire if it is analytic at every .
- Every analytic function has a Laurent series, which is like a Taylor series, but negative powers are allowed.
- If is a Laurent series in some neighbourhood of then is analytic, and is called a pole of of order . The coefficient of is the residue of at , which we shall denote as .
- If is analytic except for a set of isolated poles, it is called meromorphic.
Unlike real analysis, amazing things happen when analysis is done in the complex plane. Some important results in complex analysis are:
- Cauchy’s integral theorem, that says if is analytic on some open set and is any closed curve in , then
- Cauchy’s integral formula: If above is a circle oriented counter-clockwise and is any interior point, then
- Cauchy’s residue theorem: If is the set of poles of inside the closed curve and is analytic inside except for these poles, then
- Liouville’s theorem: Any entire function that is bounded must be constant.
- Identity theorem: If are (connected) open sets with analytic on , such that on , then on .
The identity theorem gives a useful method of extending a function analytically. Suppose are open sets and and are analytic. If on , and we were to define on preserving analyticity, then the identity theorem says on . in this case is called the (unique) analytic continuation of to . Now let’s go back to the Riemann zeta function. It’s proper definition is the following:
Recall the gamma function
It can be shown using integration by parts that . The function is analytic for , so this equation provides the analytic continuation of to . By repeating this procedure we obtain the analytic continuation of to the whole complex plane (except for the non-positive integers, which are in fact the poles of ).
It is an exercise to show that
It is another exercise to show that
where is the Hankel contour:
Now the right side of the above equation is analytic for . Hence this equation provides the analytic continuation of to (in fact, there is a pole at .)
The Bernoulli numbers are defined by the equation
and for . Substituting this into the equation above, setting and using we obtain
By Cauchy’s residue theorem,
Thus , which is zero if is even.