Number Theory: An Introduction to Mathematics

2 Chebyshev’s Functions 367

for complex values ofs. By developing these ideas, and by showing especially that
ζ(s)has no zeros on the lineRs=1, Hadamard and de la Vall ́ee Poussin proved
the prime number theorem (independently) in 1896. Shortly afterwards de la Vall ́ee
Poussin (1899) confirmed thatLi(x)was a better approximation thanx/logxtoπ(x)
by proving (in particular) that

π(x)=Li(x)+O(x/logαx) for everyα> 0. (2)

Better error bounds than de la Vall ́ee Poussin’s have since been obtained, but they still
fall far short of what is believed to be true.
Another approach to the prime number theorem was found by Wiener (1927–
1933), as an application of his general theory of Tauberian theorems. A convenient
form for this application was given by Ikehara (1931), and Bochner (1933) showed
that in this case Wiener’s general theory could be avoided.
It came as a great surprise to the mathematical community when in 1949 Selberg,
assisted by Erd ̋os, found a new proof of the prime number theorem which uses only
the simplest facts of real analysis. Though elementary in a technical sense, this proof
was still quite complicated. As a result of several subsequent simplifications it can
now be given quite a clear and simple form. Nevertheless the Wiener–Ikehara proof
will be presented here on account of its greater versatility. Theerror bound (2) can be
obtained by both the Wiener and Selberg approaches, in the latter case at the cost of
considerable complication.

2 Chebyshev’sFunctions.......................................

In his second paper Chebyshev introduced two functions

θ(x)=

∑

p≤x

logp,ψ(x)=

∑

pα≤x

logp,

which have since played a major role. Althoughψ(x)has the most complicated
definition, it is easier to treat analytically than eitherθ(x)orπ(x). As we will show,
the asymptotic behaviour ofθ(x)is essentially the same as that ofψ(x),andthe
asymptotic behaviour ofπ(x)may be deduced without difficulty from that ofθ(x).
Evidently

θ(x)=ψ(x)=0forx< 2

and

0 <θ(x)≤ψ(x) forx≥ 2.

Lemma 3The asymptotic behaviours ofψ(x)andθ(x)are connected by

(i)ψ(x)−θ(x)=O(x^1 /^2 log^2 x);
(ii)ψ(x)= O(x)if and only ifθ(x)= O(x), and in this caseψ(x)−θ(x)=
O(x^1 /^2 logx).