In the same way, Riemann found a way of extending the above infinite series definition of the zeta-function to all numbers x other than 1 (including 0 and −1). But he went much further than this. Using a technique called ‘analytic continuation’, Riemann extended the definition of the zeta function to all complex numbers other than 1 (because ζ(1) is undefined) in such a way that when k is a real number greater than 1, we get the same value as before. Because of this, the function is now known as the Riemann zeta function.
In Chapter 7 we saw Gauss’s attempt to explain why the primes thin out on average, by proposing the estimate x/log x for the number of primes up to x. Riemann’s great achievement was to obtain an exact formula for the number of primes up to x, and his formula involved in a crucial way the so-called zeros of the zeta function—that is, the complex numbers z that satisfy the equation . But where are these zeros?
It turns out that when ; these are called the trivial zeros of the zeta function. All the other zeros of the zeta function, the non-trivial zeros, are known to lie within a vertical strip between and (the so-called critical strip), as shown in Figure 36. As we move away from the horizontal axis, the first few non-trivial zeros occur at the following points:
36. The zeros of the Riemann zeta function in the complex plane.
Here, the imaginary parts (such as 14.1) are approximate, but the real parts are all equal to . Because all of these points all have the form multiple of i, the question arises:
Does every zero of the Riemann zeta function in the critical strip lie on the line x= ?
The Riemann hypothesis is that the answer to this question is ‘yes’. It has been proved that the zeros in the critical strip are symmetrically placed, both above and below the x-axis and on either side of the line , and that as we progress vertically up and down the line , many zeros do lie on it—in fact, the first trillion zeros lie on this line! But do all of the non-trivial zeros lie on the line , or might the first trillion be just a coincidence?
It’s now generally believed that all the non-trivial zeros do lie on this line, but proving this is one of the most difficult unsolved challenges throughout the whole of mathematics. Indeed no-one has yet been able to prove the Riemann hypothesis, even after a century and a half. The million dollar prize is still up for grabs!
Consequences
If the actual statement of the Riemann hypothesis seems an anticlimax after the big build-up, its consequences are substantial. Recalling Riemann’s discovery of the role that the zeta function’s zeros play in the prime-counting function π(x) and in his exact formula (involving the zeta function) for the number of primes up to x, we note that any divergence of these zeros from the line would crucially affect Riemann’s exact formula, because our understanding about how the prime numbers behave is so tied up in this formula. Indeed, finding just one zero off the line would cause major havoc in number theory—and in fact throughout mathematics: for a mathematician, truth must be absolute, and admitting even a single exception is forbidden. The prime number theorem would still be true, but would lose its influence on the primes. Instead of Don Zagier’s ‘military precision’, mentioned in Chapter 7, the primes would be found to be in full mutiny!
We conclude this chapter with an unexpected development. In 1972 the American number theorist Hugh Montgomery was visiting the tea-room at Princeton’s Institute for Advanced Study and found himself sitting opposite the celebrated physicist Freeman Dyson. Montgomery had been exploring the spacings between the zeros on the critical line, and Dyson said ‘But those are just the spacings between the energy levels of a quantum chaotic system’. If this analogy indeed holds, as many think possible, then the Riemann hypothesis may well have consequences in quantum physics. Conversely, using their knowledge of these energy levels, quantum physicists rather than mathematicians may be the ones to prove the Riemann hypothesis. This is truly an intriguing thought!
Chapter 9Aftermath
In the previous seven chapters we have explored a range of topics from number theory, and we can now return to the questions that we raised in Chapter 1. Most of these questions we were able to answer, while a few others turned out to be famous problems that remain unresolved. For each question we refer back to the chapter in which it was discussed.
The first ten questions
We opened Chapter 1 by posing ten questions.
In which years does February have five Sundays?
Questions that ask for the day of the week on which a particular event falls have a long history, and their solutions often involve the idea of congruence. As we saw in Chapter 4, February has five Sundays in the years 2004, 2032, 2060, and 2088. Also, because 2000 was a leap year, we can find the corresponding years in the previous century by subtracting multiples of 28 years from 2004; those years were 1976, 1948, and 1920. The corresponding years for other centuries can be found in a similar way.
What is special about the number 4,294,967,297?
This number is . As we saw in Chapter 3, Pierre de Fermat believed that all numbers of the form , where n is a power of 2, are prime (the so-called ‘Fermat primes’), and this number, where , was the smallest that
