wHITTy & wIlSON | 401
define ζ(k) for other numbers k? For example, how might we define ζ(0) or ζ(−1)? We cannot
define them by the same infinite series since we would then have
ζ() 01 =+/1^00 1/21++/3^00 1/4 ...+= 1111 ++++...
and
ζ()−= 11 ++1/21−−^11 /3 ++1/4−^1 ...=+1 234+++...,
and neither of these series converges. So we need to find some other way.
As a clue to how we proceed, we can show that, for certain values of x,
1 ++xx^23 ++xx...=1/ 1–().
We have already seen that this is true when x = 1/p. 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 is not defined). So we
can extend the definition of the series on the left-hand side to all values of x (other than 1) by
redefining it using the formula on the right-hand side.
In the same way, Riemann found a way of extending the infinite series definition of the zeta
function to all numbers x other than 1 (including 0 and –1). But he went further than this,
extending the definition to almost all complex numbers. Here, a ‘complex number’ is a symbol
of the form x + iy, where i represents the ‘imaginary square root of –1’; x is called the real part
of the complex number, and y is the imaginary part. Examples of complex numbers are 3 + 4i
Box 36.1 The zeta function and prime numbers
We can write the series for ζ(1) as follows:
(1) =1 +^1
2
+^1
3
+^1
4
+ =^1
2
+^1
4
1 + 1 +1^1 +
3
+^1
9
1
5
+^1
25
ζ... +.. .×× +.. . +.. ..×.. ,
where each bracket involves the powers of just one prime number. Summing the separate series, using
our earlier result that 1 + 1/p + 1/p^2 + . . . = p/(p – 1), gives
ζ×(1)=23/2 5/4.××..=2××11/2 11/4×....
If there were only a finite number of primes, then the right-hand side would have a fixed value.
This would mean that ζ(1) has this same value, which is not the case since it is not defined. So there
must be infinitely many primes.
Euler extended these ideas to prove that, for any number k > 1:
ζ(k) = 2k/(2k – 1) × 3 k/(3k – 1) × 5 k/(5k – 1) × 7 k/(7k – 1) × . . . .
This remarkable result provides an unexpected link between the zeta function, which involves adding
reciprocals of powers of numbers and seemingly has nothing to do with primes, and a product that
intimately involves the prime numbers. It was a major breakthrough.