Number Theory: An Introduction to Mathematics

4 The Riemann Hypothesis 381

∫∞

0

xs/^2 −^1 e−n

(^2) πx
dx=π−s/^2 Γ(s/ 2 )n−s.
Hence, ifσ>1,
Z(s)=

∑∞

n= 1

∫∞

0

xs/^2 −^1 e−n

(^2) πx
dx

=

∫∞

0

xs/^2 −^1 φ(x)dx,

where

φ(x)=

∑∞

n= 1

e−n

(^2) πx
.
By Proposition 10,
2 φ(x)+ 1 =x−^1 /^2 [2φ( 1 /x)+1].
Hence
Z(s)=

∫∞

1

xs/^2 −^1 φ(x)dx+

∫ 1

0

xs/^2 −^1 {x−^1 /^2 φ( 1 /x)+( 1 / 2 )x−^1 /^2 − 1 / 2 }dx

=

∫∞

1

xs/^2 −^1 φ(x)dx+

∫ 1

0

xs/^2 −^3 /^2 φ( 1 /x)dx+ 1 /(s− 1 )− 1 /s

=

∫∞

1

(xs/^2 −^1 +x−s/^2 −^1 /^2 )φ(x)dx+ 1 /s(s− 1 ).

The integral on the right is convergent for allsand thus provides the analytic continu-
ation ofZ(s)to the whole plane. Moreover the right side is unchanged ifsis replaced
by 1−s.

The functionZ(s)in Proposition 11 is occasionally called thecompletedzeta
function. In its product representation

Z(s)=π−s/^2 Γ(s/ 2 )Πp( 1 −p−s)−^1

it makes sense to regardπ−s/^2 Γ(s/ 2 )as an Euler factor at∞, complementing the
Euler factors( 1 −p−s)−^1 at the primesp.
It follows from Proposition 11 and the previously stated properties of the gamma
function that the definition ofζ(s)may be extended to the whole complex plane, so
thatζ(s)− 1 /(s− 1 )is everywhere holomorphic andζ(s)=0ifs=− 2 ,− 4 ,− 6 ,....
Sinceζ(s)=0forσ≥1andζ( 0 )=− 1 /2, the functional equation shows that these
‘trivial’ zeros ofζ(s)are its only zeros in the half-planeσ≤0. Hence all ‘nontrivial’
zeros ofζ(s)lie in the strip 0<σ<1 and are symmetrically situated with respect
to the lineσ= 1 /2. The famousRiemann hypothesisasserts that all zeros in this strip
actually lie on the lineσ= 1 /2.