Number Theory: An Introduction to Mathematics

370 IX The Number of Prime Numbers

It follows from (3) 1 –(3) 2 thatR(x)=O(x/logαx)for someα>0 if and only if
Q(x)=O(x/logα+^1 x). Consequently, by Lemma 3,

ψ(x)=x+O(x/logαx) for everyα> 0

if and only if

π(x)=

∫x

2

dt/logt+O(x/logαx) for everyα> 0 ,

andπ(x) then has the asymptotic expansion

π(x)∼{ 1 +1!/logx+2!/log^2 x+···}x/logx,

the error in breaking off the series after any finite number of terms having the order of
magnitude of the first term omitted.
It follows from (3) 1 –(3) 2 also that,for a givenαsuch that 1 / 2 ≤α<1,

ψ(x)=x+O(xαlog^2 x),

if and only if

π(x)=

∫x

2

dt/logt+O(xαlogx).

The definition ofψ(x)can be put in the form

ψ(x)=

∑

n≤x

Λ(n),

where thevon Mangoldt functionΛ(n)is defined by

Λ(n)=logpifn=pαfor some primepand someα> 0 ,

=0otherwise.

For any positive integernwe have

logn=

∑

d|n

Λ(d), (4)

since ifn=pα 11 ···pαssis the factorization ofninto powers of distinct primes, then

logn=

∑s

j= 1

αjlogpj.

3 ProofofthePrimeNumberTheorem

TheRiemann zeta-functionis defined by

ζ(s)=

∑∞

n= 1

1 /ns. (5)