Number Theory: An Introduction to Mathematics

4 The Riemann Hypothesis 379

The solution of this first order linear differential equation is

g(u)=g( 0 )e−πu

(^2) /y
.
Moreover
g( 0 )=

∫∞

−∞

e−v

(^2) πy
dv=(πy)−^1 /^2 J,
where

J=

∫∞

−∞

e−v

2

dv.

Thus we have proved that

∑∞

n=−∞

e−(v+n)

(^2) πy
=(πy)−^1 /^2 J

∑∞

n=−∞

e−n

(^2) π/y
e^2 πinv.
Substitutingv=0,y=1, we obtainJ=π^1 /^2.

Thetheta function
θ(x)=

∑∞

n=−∞

e−n

(^2) πx
(x> 0 )
arises not only in the theory of elliptic functions, as we will see in Chapter XII, but
also in problems of heat conduction and statistical mechanics. The transformation law
θ(x)=x−^1 /^2 θ( 1 /x)
is very useful for computational purposes since, whenxis small, the series forθ(x)
converges extremely slowly but the series forθ( 1 /x)converges extremely rapidly.
Since the functional equation of Riemann’s zeta function involves Euler’sgamma
function, we summarize here the main properties of the latter. Euler (1729) defined his
functionΓ(z)by
1 /Γ(z)=nlim→∞z(z+ 1 )···(z+n)/n!nz,
wherenz =exp(zlogn)and the limit exists for everyz ∈C. It follows from the
definition that 1/Γ(z)is everywhere holomorphic and that its only zeros are simple
zeros at the pointsz= 0 ,− 1 ,− 2 ,....MoreoverΓ( 1 )=1and
Γ(z+ 1 )=zΓ(z).
HenceΓ(n+ 1 )=n! for any positive integern. By puttingΓ(z+ 1 )=z! the definition
of the factorial function may be extended to anyz∈Cwhich is not a negative integer.
Wielandt (1939) has characterizedΓ(z)as the only solution of the functional equation
F(z+ 1 )=zF(z)