402 | 36 INTRODUCING TURING’S mATHEmATICS
and 2i (which equals 0 + 2i), and a real number such as 3 can be thought of as 3 + 0i. There is
nothing particularly mystical about this—all it means is that whenever we meet i^2 in calcula-
tions, we replace it by –1. Using a technique called ‘analytic continuation’, Riemann extended
the definition of the zeta function to all complex numbers other than 1 (since ζ(1) is unde-
fined). When k is a real number greater than 1, we get the same values as before; for example,
ζ(2) = π^2 /6.
We can represent complex numbers geometrically on the ‘complex plane’. This two-
dimensional array consists of all points (x, y), where the point (x, y) represents the complex
number x + iy; for example, the points (3, 4), (0, 2), and (1, 0) represent the complex numbers
3 + 4i, 2i, and 1.
We have seen how Gauss attempted to explain why the primes thin out by proposing the
estimate x/ln 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’—the complex numbers z that satisfy the equation ζ(z) = 0.
It turns out that ζ(z) = 0 when z = –2, –4, –6, –8, . . . (these are called the ‘trivial zeros’ of the
zeta function), and that all the other zeros of the zeta function (the ‘non-trivial zeros’) lie within
a vertical strip between x = 0 and x = 1 (the so-called ‘critical strip’). As we move away from the
horizontal axis, the first few non-trivial zeros are at the following points:
(^1) iiii
2
14.1,^1
2
21.01 ,^1
2
25.01 ,and^1
2
±±±±30.4.
The imaginary parts (such as 14.1) are approximate, but the real parts are all equal to
1
2
(Fig. 36.5). Since all of these points all have the form^12 + (something times i), the question
arises: Do all of the zeros in the critical strip lie on the line x =^12.
The Riemann hypothesis is that the answer to this question is ‘yes’. It is known that the zeros
in the critical strip are symmetrically placed, both horizontally about the x-axis and vertically
–4 –2^01
zeros at –2,–4,–6, ...
critical line
critical strip
(^1) / 2 – 14.1i
(^1) / 2 + 14.1i
(^1) / 2
(^1) / 2 – 21.01i
(^1) / 2 + 21.01i
(^1) / 2 – 25.01i
(^1) / 2 + 25.01i
figure 36.5 The zeros of the zeta function in the complex plane.