Number Theory: An Introduction to Mathematics

3 Proof of the Prime Number Theorem 375

of the function

k(t)= 1 −|t|for|t|≤ 1 ,=0for|t|≥ 1 ,

has the properties

kˆ(u)≥0for−∞<u<∞, C:=

∫∞

−∞

kˆ(u)du<∞.

Indeed

kˆ(u)=

∫ 1

− 1

eiut( 1 −|t|)dt

= 2

∫ 1

0

( 1 −t)cosut dt

= 2 ( 1 −cosu)/u^2.

Letε,λ,ybe arbitrary positive numbers. Ifs=ε+iλt,then

λ

∫ 1

− 1

eiλtyk(t)g(s)dt=λ

∫ 1

− 1

eiλtyk(t)

∫∞

0

e−εxe−iλtx{α(x)−A}dxdt

=λ

∫∞

0

e−εx{α(x)−A}

∫ 1

− 1

eiλt(y−x)k(t)dtdx

=λ

∫∞

0

e−εxα(x)kˆ(λ(y−x))dx

−λA

∫∞

0

e−εxkˆ(λ(y−x))dx.

Whenε→+0 the left side has the limit

χ(y):=λ

∫ 1

− 1

eiλtyk(t)γ(λt)dt

and the second term on the right has the limit

λA

∫∞

0

kˆ(λ(y−x))dx.

Consequently the first term on the right also has a finite limit. It follows that

λ

∫∞

0

α(x)kˆ(λ(y−x))dx

is finite and is the limit of the first term on the right. Thus

χ(y)=λ

∫∞

0

{α(x)−A}kˆ(λ(y−x))dx

=

∫λy

−∞

{α(y−v/λ)−A}kˆ(v)dv.