Number Theory: An Introduction to Mathematics

376 IX The Number of Prime Numbers

By the ‘Riemann–Lebesgue lemma’,χ(y)→0asy→∞. In fact this may be
proved in the following way. We have

χ(y)=

∫∞

−∞

eiλtyω(t)dt

where

ω(t)=λk(t)γ(λt).

Changing the variable of integration tot+π/λy, we obtain

χ(y)=−

∫∞

−∞

eiλtyω(t+π/λy)dt.

Hence

2 χ(y)=

∫∞

−∞

eiλty{ω(t)−ω(t+π/λy)}dt

and

2 |χ(y)|≤

∫∞

−∞

|ω(t)−ω(t+π/λy)|dt.

Sinceω(t)is continuous and vanishes outside a finite interval, it follows that
χ(y)→0asy→∞.
Since
∫λy

−∞

kˆ(v)dv→C asy→∞,

we deduce that

lim

y→∞

=

∫λy

−∞

α(y−v/λ)kˆ(v)dv=AC for everyλ> 0.

We now make use of the fact thatehxα(x)is a nondecreasing function. Choose any
δ∈( 0 , 1 ).Ify=x+δ,wherex≥0, then for|v|≤λδ

α(y−v/λ)≥e−h(δ−v/λ)α(x)≥e−^2 hδα(x)

and hence
∫λy

−∞

α(y−v/λ)kˆ(v)dv≥e−^2 hδα(x)

∫λδ

−λδ

kˆ(v)dv.

We can chooseλ=λ(δ)so large that the integral on the right exceeds( 1 −δ)C. Then,
lettingx→∞we obtain

AC≥e−^2 hδ( 1 −δ)C lim

x→∞

α(x).