- NUMBER SYSTEMS 197
Chebyshev in his 1859 paper on ð(χ), even though he was in close contact with
Dirichlet, and Chebyshev's paper had been published in a French journal.
The full proof of the prime number theorem turned out to involve the use
of complex analysis. As mentioned above, Riemann had studied the zeros of the
Riemann zeta function. This function was also studied by two long-lived twentieth-
century mathematicians, the Belgian Charles de la Vallee Poussin (1866-1962) and
the Frenchman Jacques Hadamard (1865-1963), who showed independently of each
other (Hadamard, 1896; Vallee Poussin, 1896) that the Riemann zeta function has
no zeros with real part equal to l.^9 Vallee Poussin showed later (1899) that
where for some á > 0 the error term en is bounded by a multiple of ne~avlogn.
Number theory did not slow down or stop after the proof of the prime num-
ber theorem. On the contrary, it exploded into a huge number of subfields, each
producing a prodigious amount of new knowledge year by year. However, we must
stop writing on this subject sometime and move on to other topics, and so we shall
close our account of number theory at this point.
To the ancient mathematicians in the Middle East and Europe, numbers meant
positive integers or ratios of them, in other words, what we call rational numbers.
In India and China negative numbers were recognized, and 0 was recognized as a
number in its own right, as opposed to merely an absence of numbers, at a very
early stage. Those numbers reached Europe only a brief while before algebra led
to the consideration of imaginary numbers. In this section we explore the gradual
expansion of the concept of a number to include not only negative and imaginary
numbers, which at least had the merit of being understandable in finite terms, but
also irrational roots of equations and transcendental real numbers such as ð and
e, and the infinite cardinal and ordinal numbers mathematicians routinely speak
about today. It is a story of the gradual enlargement of the human imagination
and the clarification of vague, intuitive ideas.
2.1. Negative numbers and zero. It was mentioned in Chapter 7 that place-
value systems of writing numbers were invented in Mesopotamia, India, China, and
Mesoamerica. What is known about the Maya system has already been described
in Chapter 5. We do not know how or even if they performed multiplication or
division or how they worked with fractions. Thus, for this case all we know is that
they had a place-value system and that it included a zero to occupy empty places.
The Mesopotamian system was sexagesimal and had no zero for at least the first
1000 years of its existence. The other three systems were decimal, and they too
were rather late in acquiring the zero. Strange though it may seem to one who has
a modern education, in India and China negative numbers seem to have been used
before zero was invented.
(^9) The fact that æ(æ) has no zeros with real part equal to 1 is an elementary theorem (see Ivic, 1985,
pp. 7 8). That does not make the prime number theorem trivial, however, since the equivalence
between this result and the prime number theorem is very difficult to prove. A discussion of the
reasons why the two are equivalent was given by Norbert Wiener; see his paper "Some prime-
number consequences of the Ikehara theorem," in his Collected Works, Vol. 2, pp. 254-257.