The History of Mathematics: A Brief Course

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  1. MODERN NUMBER THEORY 193


until Dirichlet restored its simplicity by going back very nearly to
Legendre's original construction.

In attempting to extend the law of quadratic reciprocity to higher powers,
Gauss was led to consider what are now called the Gaussian integers, that is,
complex numbers of the form m + ç í'—ú· Gauss showed that the concepts of prime
and composite number make sense in this context just as in the ordinary integers
and that every such number has a unique representation (up to multiplication by
the units ±1 and ±\f—T) as a product of irreducible factors. Notice that no prime
integer of the form An + 1 can be "prime" in this context, since it is a sum of two
squares: in + 1 = p^2 + q^2 = (p + q\/—l)(p — qy/—\). The generalization of the
notion of prime number to the Gaussian integers is an early example of the endless
generalization and abstraction that characterizes modern mathematics.
Gauss also gave an estimate of the number of primes not larger than x, in the
form of the integral

Here, as above, the logarithm means the natural logarithm. He did not, however,
prove that this approximation is asymptotically good, that is, that 7r(:r;)/Li (.x)
tends to 1 as ÷ tends to infinity. That is the content of the prime number theorem.

1.6. Dirichlet. The works of Gauss on number theory were read by another bright
star of nineteenth-century mathematics, Johann Peter Gustav Lejeune-Dirichlet
(1805-1859), who contributed several gems to this difficult area. He was of Belgian
ancestry (hence his French-sounding name, even though he was a German). He was
born in the city of Duren, which lies between Aachen (Aix) and Koln (Cologne),
but went to Paris to study at the age of 16. At the age of 20 he proved the case
ç = 5 of Fermat's last theorem. (Legendre, who was the referee for Dirichlet's
paper, contributed his own proof of one subcase of this case.) That same year he
returned to Germany and took up a position at the University of Breslau. In 1828
he went to Berlin and was the first star in a bright galaxy of Berlin mathematicians.
In 1831 he was elected to the Berlin Academy of Sciences. That year he married
Rebekah Mendelssohn, sister of the composers Felix and Fanny Mendelssohn. In
1855, dissatisfied with the heavy teaching loads in Berlin, he moved to Gottingen as
the successor of Gauss, who had died that year. In 1858 he suffered a heart attack
and the death of his wife, and in 1859 he himself succumbed to heart disease.
Although Dirichlet also worked in the theory of Fourier series and analytic
function theory, having given the first rigorous discussion of the convergence of a
Fourier series in 1829 and the modern definition of a function in 1837, we are at
the moment concerned with his contributions to number theory. One of these is his
1837 theorem, already mentioned, that each arithmetic sequence in which the first
term and the common difference are relatively prime contains an infinite number
of primes. To prove this result, he introduced what is now called the Dirichlet
character χ(ç) = (-1)* if ç = 2k + 1, χ(ç) = 0 if ç is even, along with the
Dirichlet series


This work brought number theory and analysis together in the subject now
called analytic number theory. According to Weil (1984, PP- 252-256), the two


n=l
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