Number Theory: A Very Short Introduction

of the series (see Figure 34), we can get as close to 2 as we wish by adding enough of them—say, the first hundred or the first million: for example, we can get within 0.001 of 2 by adding the first twelve terms. We express this by saying that the infinite series converges to 2, or has the finite sum 2, and we write

34. Summing the powers of

In the same way, we can show that the infinite series

whose denominators are the powers of 3, converges to , and that the infinite series

whose denominators are the powers of 5, converges to , More generally, we can show that, for any number p,

We’ll need this result later on.

Not all infinite series converge. A celebrated example is the so-called harmonic series

whose denominators are the positive integers. To see why it doesn’t have a finite sum we’ll group the terms as

Now this sum is larger than the following sum:

which equals  , because each group of terms has sum .

But the sum of this latter series increases without limit as we add more terms, and the harmonic series has an even larger sum and so it cannot have a finite sum either.

So the harmonic series doesn’t converge: and surprisingly, as Euler proved in 1737, even if we throw away most of its terms and leave only those whose denominators are primes—that is,

—then there is still no finite sum.

The zeta function

In the early 18th century a celebrated challenge was to find the exact sum of the infinite series

whose denominators are the squares, 1, 4, 9, 16, 25, … . The Swiss mathematician Johann Bernoulli, then possibly the world’s greatest mathematician, failed to solve the problem, and it was eventually answered by his former pupil, Leonhard Euler, who proved that this series converges to π²/6, a remarkable result in that it involves the ‘circle number’ π. As Euler proudly observed:

Quite unexpectedly I have found an elegant formula involving the quadrature of the circle.

In the same way, Euler proved that:

when the denominators are the fourth powers, the sum is π⁴/90,

when the denominators are the sixth powers, the sum is π⁶/945,

and when the denominators are the eighth powers, the sum is π⁸/9450.

He subsequently continued his calculations up to the twenty-sixth powers. Here the sum is

which he calculated correctly.

When the denominators are the nth powers, Euler denoted the sum as ζ(n), where ζ is the Greek letter zeta, and named it the zeta function—that is,

So ζ(1) is undefined (because the harmonic series has no finite sum), but , , , etc. It turns out that the series for ζ(n) converges for every number n that is greater than 1.

Although the zeta function ζ(n) may seem to have nothing in common with prime numbers, Euler spotted a crucial connection, which we now explore. This connection can be used to give another proof that the list of primes is never-ending.The zeta function and prime numbers

We can write the series for ζ(1) as follows:

where each bracket involves the powers of just one prime number. This is because, by unique factorization, each term 1/n in the series for ζ(1) appears exactly once in the product on the right: for example,

We now sum the series in each bracket, using our earlier result that

this gives

We can use this product to prove that there are infinitely many primes. For, if there were only finitely many primes, then the right-hand side would be a finite product, and so would have a fixed value. But this would require ζ(1) to have this same value, which is impossible because it is undefined. So there must be infinitely many primes.

Euler extended these ideas to prove that, for any number n that’s greater than 1,

This remarkable result is called the Euler product, and provides an unexpected link between the zeta function, which involves powers of numbers and seems to have nothing to do with primes, and a product that intimately involves all the prime numbers. It was a major breakthrough.Complex numbers

Before we present the Riemann hypothesis, we’ll also need the idea of a complex number. This involves i, the ‘imaginary’ square root of −1, which we met briefly in Chapter 7.A complex number is a symbol of the form ; x is called the real part of the complex number, and y is the imaginary part. Examples of complex numbers are (which equals ), and 3 (which can be thought of as ).

We can represent complex numbers geometrically as points on the ‘complex plane’. This two-dimensional picture consists of all points (x, y), where (x, y) represents the complex number ; for example, the points , and (3, 0) represent the complex numbers , and 3 (see Figure 35).

35. Points on the complex plane.

The Riemann hypothesis

We’ve now set the scene for the Riemann hypothesis.

As we’ve seen, the zeta function ζ(n) is defined for any number n that is greater than 1. But can we define it for other numbers n too? For example, how might we define ζ(0) or ζ(−1)? We can’t define them by the same infinite series, because we’d then have

and

and neither of these series has a finite sum. So we’ll need to find some other way.

As a clue to how to proceed, we can show that, for certain values of x,

we’ve already seen that this is true when . But it can be shown that the series on the left-hand side converges only when x lies between −1 and 1, whereas the formula on the right-hand side has a value for any x, apart from 1 (when we get 1/0, which