One of the supreme
achievements of 19th-century mathematics was the prime number theorem, and
it is worth a brief digression. To begin, designate the number of primes less
than or equal to n by π(n). Thus π(10) = 4
because 2, 3, 5, and 7 are the four primes not exceeding 10. Similarly
π(25) = 9 and π(100) = 25. Next, consider the proportion of
numbers less than or equal to n that are prime—i.e., π (n)/n.
Clearly π (10)/10 = 0.40, meaning that 40 percent of the numbers not
exceeding 10 are prime. Other proportions are shown in the table.
A pattern is anything
but clear, but the prime number theorem identifies one, at least approximately,
and thereby provides a rule for the distribution of primes among the whole
numbers. The theorem says that, for large n, the proportion π(n)/n is
roughly 1/log n, where log n is the natural logarithm of n. This link between primes
and logs is nothing short of extraordinary.
One of the first to
perceive this was the young Gauss, whose examination of log tables and prime
numbers suggested it to his fertile mind. Following Dirichlet’s exploitation of
analytic techniques in number theory, Bernhard Riemann (1826–66) and Pafnuty
Chebyshev (1821–94) made substantial progress before the prime number
theorem was proved in 1896 by Jacques Hadamard (1865–1963) and
Charles Jean de la Vallée-Poussin (1866–1962). This brought the 19th century to
a triumphant close.
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