analytic continuation of Riemann zeta to critical strip

analytic continuation of Riemann zeta to critical strip

The

1

n

s

=

e

-

s

⁢

log

⁡

n

(see the

general power

) of the series

∑

n

=

1

∞

1

n

s

=

1

+

1

2

s

+

1

3

s

+

1

4

s

+

…

,

(1)

defining the

Riemann zeta function

ζ

⁢

(

s

)

for

ℜ

⁡

s

>

1

,  are

holomorphic

in the whole

s

-plane and the
series

converges uniformly

in any closed disc of the
half-plane

ℜ

⁡

s

>

1

((let

s

=

σ

+

i

⁢

t

with

σ

,

t

∈

ℝ

and

σ

>

1

;
then

|

1

n

s

|

=

1

n

σ

≤

1

n

1

+

d

for a

positive

d

for all

n

=

1

,

2

,

…

;
the series

∑

n

=

1

∞

1

n

1

+

d

converges

since

1

+

d

>

1

;  thus the series (1) converges uniformly
in the closed half-plane

ℜ

⁡

s

≥

1

+

d

,  by the
Weierstrass criterion (

http://planetmath.org/WeierstrassCriterionOfUniformConvergence

))).
Therefore we can infer (see

theorems on complex function series

(

http://planetmath.org/TheoremsOnComplexFunctionSeries

))
that the sum

ζ

⁢

(

s

)

of (1) is holomorphic in the domain

ℜ

⁡

s

>

1

.

We use also the fact that the series

∑

n

=

1

∞

(

-

1

)

n

-

1

n

s

=

1

-

1

2

s

+

1

3

s

-

1

4

s

+

-

…

(2)

defining the

Dirichlet eta function

η

⁢

(

s

)

, a.k.a. the

alternating zeta function

, is

convergent

for

ℜ

⁡

s

>

0

and its sum is holomorphic in this half-plane.

If we multiply the series (1) by the

difference

1

-

2

2

s

, every other of the series changes its sign and we get the series (2).  So we can write

ζ

⁢

(

s

)

=

η

⁢

(

s

)

1

-

2

2

s

,

(3)

which is valid when the

denominator

does not vanish and

ℜ

⁡

s

>

1

.  The zeros of the denominator are obtained from

2

s

=

2

,  i.e. from

e

s

⁢

log

⁡

2

=

e

log

⁡

2

.

This gives

s

⁢

log

⁡

2

-

log

⁡

2

=

n

⋅

2

⁢

i

⁢

π

(see the

periodicity

of

exponential function

), i.e.

s

=

1

+

n

⋅

2

⁢

π

⁢

i

ln

⁡

2

(

n

∈

ℤ

)

.

(4)

Thus the zeros of the denominator of (3) are on the line

ℜ

⁡

s

=

1

.

Now the

function

on the right hand side of (3) is holomorphic in the set

D

:=

{

s

∈

ℂ

⋮

ℜ

s

>

0

}

∖

{

1

+

n

⋅

2

⁢

π

⁢

i

ln

⁡

2

⋮

n

∈

ℤ

}

and the values of this function coincide with the values of

zeta function

in the half-plane

ℜ

⁡

s

>

1

.

This result means that, via the equation (3), the zeta function has been analytically continued (

http://planetmath.org/AnalyticContinuation

) to the domain

D

, as far as to the

imaginary axis

.

Remark.

In reality, all points (4) except

s

=

1

are

removable singularities

of

ζ

⁢

(

s

)

given by (3), due to the fact that they are also zeros of

η

⁢

(

s

)

.  The fact is considered in the entry

zeros of Dirichlet eta function

.

Charles Hermite has shown that the zeta function may be analytically continued to the whole

s

-plane except for a

simple pole

at

s

=

1

,  by using the equation

ζ

⁢

(

s

)

=

1

Γ

⁢

(

s

)

⁢

∫

0

∞

x

s

-

1

e

x

-

1

⁢

𝑑

x

.

(5)

See this article (

http://planetmath.org/AnalyticContinuationOfRiemannZetaUsingIntegral

).

Title

analytic continuation

of Riemann zeta to

critical strip

Canonical name

AnalyticContinuationOfRiemannZetaToCriticalStrip

Date of creation

2015-08-22 13:33:33

Last modified on

2015-08-22 13:33:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

23

Author

pahio (2872)

Entry type

Example

Classification

msc 30D30

Classification

msc 30B40

Classification

msc 11M41

Related topic

RiemannZetaFunction

Related topic

AnalyticContinuation

Related topic

MeromorphicExtension

Related topic

CriticalStrip

Related topic

AnalyticContinuationOfRiemannZetaUsingIntegral

Related topic

FormulaeForZetaInTheCriticalStrip

Related topic

GammaFunction

Defines

alternating zeta function

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