Number Theory: An Introduction to Mathematics

374 IX The Number of Prime Numbers

is defined forRs>h, where h> 0. If there exists a constant A and a function G(s),
which is continuous in the closed half-planeRs≥h, such that

G(s)=F(s)−Ah/(s−h) forRs>h,

then

φ(x)∼Aehx for x→+∞.

Proof For eachX>0wehave

∫ X

0

e−sxdφ(x)=e−sX{φ(X)−φ( 0 )}+s

∫ X

0

e−sx{φ(x)−φ( 0 )}dx.

For reals=ρ>hboth terms on the right are nonnegative and the integral on the left
has a finite limit asX→∞. Hencee−ρXφ(X)is a bounded function ofXfor each
ρ>h. It follows that ifRs>hwe can letX→∞in the last displayed equation,
obtaining

F(s)=s

∫∞

0

e−sx{φ(x)−φ( 0 )}dx forRs>h.

Hence

[G(s)−A]/s=F(s)/s−A/(s−h)=

∫∞

0

e−(s−h)x{α(x)−A}dx,

whereα(x)=e−hx{φ(x)−φ( 0 )}. Thus we will prove the theorem if we prove the
following statement:
Letα(x)be a nonnegative function for x≥ 0 such that

g(s)=

∫∞

0

e−sx{α(x)−A}dx,

where s=σ+i t , is defined for everyσ> 0 and the limit

γ(t)= lim

σ→+ 0

g(s)

exists uniformly on any finite interval−T≤t≤T. If, for some h> 0 ,ehxα(x)is a
nondecreasing function, then

lim

x→∞

α(x)=A.

In the proof of this statement we will use the fact that the Fourier transform

kˆ(u)=

∫∞

−∞

eiutk(t)dt