Number Theory: An Introduction to Mathematics

3 Proof of the Prime Number Theorem 371

This infinite series had already been considered by Euler, Dirichlet and Chebyshev,
but Riemann was the first to study it for complex values ofs. As customary, we write
s=σ+it,whereσandtare real, andn−sis defined for complex values ofsby

n−s=e−slogn=n−σ(cos(tlogn)−isin(tlogn)).
To show that the series (5) converges in the half-planeσ>1 we compare as
in§1 the sum with an integral. Ifxdenotes again the greatest integer≤x,thenon
integrating by parts we obtain

∫N

1

x−sdx−

∑N

n= 1

n−s=

∫ N+

1 −

x−sd{x−x}

=− 1 +s

∫N

1

x−s−^1 {x−x}dx.

Since
∫ N

1

x−sdx=( 1 −N^1 −s)/(s− 1 ),

by lettingN→∞we see thatζ(s)is defined forσ>1and

ζ(s)= 1 /(s− 1 )+ 1 −s

∫∞

1

x−s−^1 {x−x}dx.

But, sincex−xis bounded, the integral on the right is uniformly convergent in any
half-planeσ≥δ>0. It follows that the definition ofζ(s)can be extended to the half-
planeσ>0, so that it is holomorphic there except for a simple pole with residue 1 at
s=1.
The connection between the zeta-function and prime numbers is provided by
Euler’s product formula, which may be viewed as an analytic version of the funda-
mental theorem of arithmetic:

Proposition 5ζ(s)=

∏

p(^1 −p

−s)− (^1) forσ> 1 , where the product is taken over all
primes p.
Proof Forσ>0wehave
( 1 −p−s)−^1 = 1 +p−s+p−^2 s+···.
Since each positive integer can be uniquely expressed as a product of powers of distinct
primes, it follows that
∏
p≤x
( 1 −p−s)−^1 =

∑

n≤Nx

n−s,

whereNxis the set of all positive integers, including 1, whose prime factors are all
≤x.ButNxcontains all positive integers≤x. Hence
∣
∣
∣
∣ζ(s)−

∏

p≤x

( 1 −p−s)−^1

∣

∣

∣

∣≤

∑

n>x

n−σ forσ> 1 ,

and the sum on the right tends to zero asx→∞.