400 | 36 INTRODUCING TURING’S mATHEmATICS
where each term is one-third of the previous one, converges to 1^12 , and that, for any number p,
we can write
11 ++/1pp/1^23 ++/pp...= /–()p 1.We shall need this last result later on.
However, not all infinite series converge. One celebrated example of a series that does not
have a finite sum is the so-called harmonic series
1 1
21
31
4++++...,where the denominators are the natural numbers 1, 2, 3, . . . . To see why this series does not
converge, we bracket it as
1 1
21
31
41
51
61
71
8++ + ....
++++
+But the sum of the numbers in each bracketed expression is greater than^12 , and so the sum of
the whole series is greater than 1+^12 +++ 21 21 ...which increases without limit. So the harmonic
series cannot converge to any finite number. And amazingly, even if we throw away most of the
terms and leave only those whose denominators are prime numbers,
1
21
31
51
71
11+++++...,then there is still no finite sum.
In the early 18th century a celebrated challenge was to find the exact sum of the infinite
series
11
41
91
161
25+++++...,where the denominators are the squares, 1, 4, 9, 16, . . . . This was answered by the Swiss mathe-
matician Leonhard Euler, who showed that this series converges to π^2 /6, an unexpected and
remarkable result. In the same way, he showed that the sum of the reciprocals of the fourth
powers is π^4 /90, for the sixth powers it is π^6 /945, and so on. He called the sum of the kth powers
ζ(k), naming it the ‘zeta function’, that is,
ζ()k=+ 11 /2kk++1/31/4k+....So ζ(1) is undefined (since the harmonic series has no finite sum), but ζ(2) = π^2 /6, ζ(4) = π^2 /90,
etc. It turns out that the series for ζ(k) converges for every number k > 1.
Although the zeta function and prime numbers do not seem to have anything in common,
Euler spotted a crucial link, which we present in Box 36.1. This link can be used to give another
proof that there are infinitely many primes.
We have now set the scene for the Riemann hypothesis, in which Alan Turing became so
interested. As we have seen, the zeta function ζ(k) is defined for any number k > 1. But can we