194 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSsubjects had been drawing closer together ever since Euler began his study of
elliptic functions. Elliptic functions have played a prominent role in number theory
since 1830. Carl Gustav Jacob Jacobi (1804-1851), whose work is discussed in
Chapter 17, published a treatise on elliptic functions in 1829 in which he used these
functions to derive a formula equivalent to(1) Ó =l+8^^-l)?2fc-I + 24^E
CT(
2^
1)^
(2fc"
1)'
\n=-oo J fc = l fc=lj = l
where ó(ô) is the sum of the divisors of r. For r > 0, it is obvious that the
coefficient of qr on the left is 16 times the number of ways in which r can be
represented as the sum of four ordered nonzero squares, plus 32 times the number
of such representations as a sum of three squares, plus 24 times the number of
representations as a sum of two squares, plus 8 if the number happens to be a
square. The coefficient of qr on the right is either eight or 24 times the sum of the
divisors of the largest odd number that divides r. Since that sum is always positive,
the four-square theorem is a consequence, but much additional information is added
on the number of such representations.
The study of Dirichlet series, in particular the simplest one of all, which defines
what is now called the Riemann zeta function
oo
*•—' Ôë'
n=l
(one of several zeta functions named after distinguished mathematicians), turned
out to be important in both complex analysis and number theory. The zeta function
was introduced, though not under that name, by Euler, who gave the formula<^2 > t±- Ð (É-£'
n = l ñ prime x
The fact that the terms in the sum are indexed by all positive integers while the
factors in the product are indexed by the prime numbers accounts for the deep
connections of this function with number theory. Its values at the even integers can
be computed in terms of the Bernoulli numbers.^6 In fact, the Bernoulli numbers
were originally introduced this way. Nowadays, the nth Bernoulli number Bn is
defined to be n! times the coefficient of xn in the Maclaurin series of x/(ex — 1).
1.7. Riemann. Another giant of nineteenth-century mathematics was Georg Bern-
hard Riemann (1826-1866), who despite his brief life managed to make major con-
tributions to real and complex analysis, geometry, algebraic topology, and mathe-
matical physics. He was inspired by Legendre's work on number theory and studied
under Gauss at Gottingen, where he also became a professor, Dirichlet's successor
after 1859. Because of his frail health (he succumbed to tuberculosis at the age
of 40), he spent considerable time in Italy, where he made the acquaintance of the
productive school of Italian geometers, including Enrico Betti (1823-1892). His
greatest contribution to number theory was to attempt a rigorous estimate of ð(ç).
For this purpose he studied the zeta function introduced above and made the fa-
mous conjecture that except for its obvious zeros at the even negative integers, all
(^6) The Bernoulli numbers were the object of the first computer program written for the Babbage
analytical engine.