The Riemann hypothesis is the statement that the zeros of a certain complex-valued function ζ(s) of a complex number s all have a certain special form. That is, if we look at ζ(s) = 0 as an equation to solve, then all solutions which are real numbers have s = -2k for integers k ≥ 1, and all other solutions (which are complex) have a “real part” equal to 1/2, or symbolically, Re(s) = 1/2. Although other mathematicians before G. F. B. Riemann (1826-66) studied the zeta function (including Euler, as we shall see), the notation is Riemann’s, and hence the function is commonly known as the zeta function, after the greek letter ζ.

David Hilbert, in his famous speech at the International Congress of Mathematicians at Paris in 1900, included this problem as number 8 in his list of 23 challenging problems for mathematicians in the coming century. Today, over 100 years later, it is one of the few on that list that have not been solved. Many contemporary mathematicians consider it the most important unsolved problem in mathematics. In any case, it has been included among the seven most important problems, for the solution of any of which a prize of $1 million has been offered by the Clay Mathematics Institute.

The definition of the function ζ(s) is not especially difficult. The statement of the Riemann hypothesis is certainly clear enough (if you have any acquaintance with complex numbers). So why on Earth is this hypothesis considered so important? One reason is that it has turned out, in spite of its apparent simplicity, to be so difficult to prove. But a more important reason is that the problem is thoroughly entangled with many questions about prime numbers – which of course are of intense interest to number theorists, yet at first glance have little to do with functions of a complex variable.

We shall soon see that there *is* a very close and simple relationship between ζ(s) and prime numbers. But hidden beneath this simple relationship seems to be one much deeper and more profound – one that connects properties of the zeros of ζ(s) to the way in which prime numbers are distributed among the integers. And not only is there reason to suspect such a relationship, but much more general relationships are suspected to exist between various generalizations of ζ(s) and more general types of algebraic objects.

The function ζ(s) is defined by the infinite sum

Where the sum is taken over integers n such that 1 ≤ n < ∞.

A great deal of care is required when working with “infinite” sums, because only in certain circumstances do they actually make sense. In this case, it can be shown that the sum actually “converges” only when the real part of s is greater than 1 (i. e. Re(s) > 1). (Recall that any complex number can be written uniquely in the form s = σ + iτ, where σ and τ are both real and i^{2}= -1. We write Re(s) = σ and Im(s) = τ to denote the real and imaginary parts, respectively.)

To say that the infinite sum “converges” for Re(s) > 1 is to say that the complex numbers represented by finite partial sums of the form

have a well-defined limiting value, for each s with Re(s) > 1, as N becomes arbitrarily large.

Without going into further details, let it suffice to say that there are standard techniques for rigorously talking about infinite sums and proving that they make good sense (i. e. “converge”) under appropriate conditions. These rigorous techniques were fully established only in the second half of the 19th century. However, mathematicians had worked with infinte sums long before that – rearranging terms and performing other kinds of formal operations (e. g. differentiation and integration) – in order to derive a wide variety of formal identities. Often these formal manipulations could be rigorously justified, at least under suitable conditions. Sometimes, unfortunately, they could not, in which case nonsense results such as 1 = 0 could be derived, just as can happen when improperly allowing division by 0. Indeed, the basic operations of calculus itself (differentiation and integration) themselves involve operations with infinite sums and sequences. It was a most welcome accomplishment of 19th century mathematics to put all such “infinite” operations on a rigorous basis.

With standard rigorous techniques for working with infinite sums (which are also often called “infinite series”) in hand, it was clear that the infinte sum used above to define ζ(s) definitely did *not* converge when Re(s) < 1. Just for example, if s = -1, it would amount to trying to assign a reasonable value to the sum 1 + 2 + 3 + … . Now, there are roundabout ways for assigning plausible values to such a sum. For instance, the remarakable Indian mathematician Srinivasa Ramanujan (1887-1920) thought that 1 + 2 + 3 + … should equal -1/12. And soon we shall see a sense in which ζ(-2k), which is formally 1 + 2^{2k} + 3^{2k} + …, has a simple numerical value for integers k ≥ 1. But all of this is beyond the standard rigorous concept of convergence.

Instead of representing functions as infinite series, mathematicians developed an entirely different way of defining functions of a complex variable. It is called the method of “analytic continuation”. It turns out that complex-valued functions of complex variables have remarkable properties. For one thing, if such a function has even a first derivative (at a given point), it must actually have derivatives of all orders (at that point). Functions with that property are called “analytic” or “holomorphic”. Although this property is specific to a given point, for all “reasonable” functions it occurs for all but (at most) an isolated set of points in the complex plane. Another surprising property of analytic functions is that if they are constant on any circular disk in the complex plane (that is, a set of the form {z : |z – z_{0}| < r}), then they must be constant everywhere in the complex plane that they are analytic. (The notation |w|, for a complex number w, is called the “norm” or “absolute value” of w and is defined as √(Re(w)^{2} + Im(w)^{2}).) Consequently, any two analytic functions that are equal on some disk – so that their difference is an analytic function equal to the constant 0 on the disk – must be equal everywhere. Using this fact, the process of analytic continuation consists of extending analytic functions defined on some disk in the complex plane to the whole plane (perhaps with a few isolated singular points) by defining the function on a sequence of overlapping disks.

Now, the infinite series used to define ζ(s) for Re(s) > 1 defines the function not merely on a disk but on an infinite half of the complex plane. It can be shown that anaytic continuation permits ζ(s) to be defined for any complex s except for isolated singularities. (s = 1 is one of these.) The zeta function is therefore well-defined and a generally quite well-behaved function for all but isolated singularities. So it can be meaningfully discussed even when its infinite series doesn’t make sense. We shall soon see that there is a “functional equation” which specifies the value of ζ(1-s) in terms of ζ(s) – in particular, for Re(s) < 0.

Although the function is usually called the “Riemann” zeta function, Riemann, as we noted, was not the first to work with it. Leonhard Euler (1707-83) studied the series 100 years eariler, discovered a number of its properties, and in fact found another extremely important representation of it in terms of an infinite product instead of an infinite series. P. G. L. Dirichlet (1805-59) (who was perhaps Riemann’s favorite teacher at the University of Göttingen) studied more general series of the form ∑_{n} a_{n}/n^{s}, for arbitrary complex coefficients a_{n}. Accordingly, series of this form are called Dirichlet series.

Riemann’s contribution, however, was to realize that ζ(s) could be analytically continued to the whole complex plane, to derive many important properties of ζ(s), such as its functional equation, and – most importantly – to suspect its deeper relationship to the distribution of prime numbers. And along the way, to state his celebrated hypothesis. So there’s plenty of justification for calling ζ(s) the Riemann zeta function.