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Table 36.1 x, p(x), and their ratio x/p(x).
x p(x) x/p(x)
10 4 2.5
100 25 4.0
1000 168 6.0
10,000 1229 8.1
100,000 9592 10.4
1,000,000 78,498 12.7
10,000,000 664,579 15.0
100,000,000 5,761,455 17.4The Riemann hypothesis
What is the Riemann hypothesis? It was introduced by Bernhard Riemann, who died at the age
of 40 while a professor at Göttingen University where he had followed in the footsteps of Gauss.
Elected to the Berlin Academy in 1859, Riemann was required to contribute to the academy’s
Proceedings, and the result was his only paper in number theory, ‘On the number of primes less
than a given magnitude’ (now regarded as a classic).^12
In broad terms the Riemann hypothesis asks: ‘Do all the solutions of a particular equation
have a particular form?’. This is very vague, and the full statement, which still remains unan-
swered 150 years later, is: Do all the non-trivial zeros of the zeta function have real part^12? But
what does this mean, and what is its connection with prime numbers?
To answer these questions we need to enter the world of infinite series. The series
1 1
21
41
81
16+++++...,where the denominators are the powers of 2, continues for ever, which is why it is called an
‘infinite series’. What happens if we add all the numbers in the series together? If we add them
up one by one, we get
1, then 11
211
2,then11
21
413
4,then11
21
41
817
8+= ,
++=
+++=
and so on. Whenever we add just a finite number of terms of the series we never actually reach
2, but by adding together enough terms we can get as close to 2 as we wish; for example, we can
get within 1/1000 of 2 by adding the first twelve terms of the series. We express this by saying
that the infinite series converges to 2, or has the finite sum 2, and write
1 1
21
41
81
16+++++=... 2.In the same way, we can show that the infinite series
1 1
31
91
27++++...,