Number Theory: An Introduction to Mathematics

380 IX The Number of Prime Numbers

withF( 1 )=1 which is holomorphic in the half-planeRz >0 and bounded for
1 <Rz<2.
It follows from the definition ofΓ(z)and the product formula for the sine function
that

Γ(z)Γ( 1 −z)=π/sinπz.

Many definite integrals may be evaluated in terms of the gamma function. By repeated
integration by parts it may be seen that, ifRz>0andn∈N,then

n!nz/z(z+ 1 )···(z+n)=

∫n

0

( 1 −t/n)ntz−^1 dt,

wheretz−^1 =exp{(z− 1 )logt}. Lettingn→∞, we obtain the integral representation

Γ(z)=

∫∞

0

e−ttz−^1 dt forRz> 0. (9)

It follows thatΓ( 1 / 2 )=π^1 /^2 ,since

∫∞

0

e−tt−^1 /^2 dt=

∫∞

−∞

e−v

2

dv=π^1 /^2 ,

by the proof of Proposition 10. It was already shown by Euler (1730) that

B(x,y):=

∫ 1

0

tx−^1 ( 1 −t)y−^1 dt=Γ(x)Γ(y)/Γ(x+y),

the relation holding forRx > 0andRy > 0. The unit ball inRnhas volume
κn:=πn/^2 /(n/ 2 )! and surface contentnκn.Stirling’s formula,n!≈(n/e)n

√

2 πn,
follows at once from the integral representation

logΓ(z)=(z− 1 / 2 )logz−z+( 1 / 2 )log 2π−

∫∞

0

(t−t− 1 / 2 )(z+t)−^1 dt,

valid for anyz∈Cwhich is not zero or a negative integer. Euler’s constant

γ= lim

n→∞

( 1 + 1 / 2 + 1 / 3 +···+ 1 /n−logn)≈ 0. 5772157

mayalsobedefinedbyγ=−Γ′( 1 ).
We now return to the Riemann zeta function.

Proposition 11The function Z(s)=π−s/^2 Γ(s/ 2 )ζ(s)satisfies the functional equa-
tion

Z(s)=Z( 1 −s)for 0 <σ< 1.

Proof From the representation (9) of the gamma function we obtain, forσ>0and
n≥1,