value of the Riemann zeta function at s=0

value of the Riemann zeta function at

s

=

0

Theorem

.

Let

ζ

denote the

meromorphic extension

of the

Riemann zeta function

to the

complex plane

. Then

ζ

⁢

(

0

)

=

-

1

2

.

Proof.

Recall that one of the for the Riemann zeta function in the

critical strip

is given by

ζ

⁢

(

s

)

=

1

s

-

1

+

1

-

s

⁢

∫

1

∞

x

-

[

x

]

x

s

+

1

⁢

𝑑

x

,

where

[

x

]

denotes the

integer part

of

x

.

Also recall the

functional equation

ζ

⁢

(

s

)

=

2

s

⁢

π

s

-

1

⁢

sin

⁡

π

⁢

s

2

⁢

Γ

⁢

(

1

-

s

)

⁢

ζ

⁢

(

1

-

s

)

,

where

Γ

denotes the

gamma function

.

The only pole (

http://planetmath.org/Pole

) of

ζ

occurs at

s

=

1

. Therefore,

ζ

is

analytic

, and thus

continuous

, at

s

=

0

.

Let

lim

s

→

0

+

denote the limit as

s

approaches

0

along any path contained in the region

Re

⁡

(

s

)

>

0

. Thus:

ζ

⁢

(

0

)

=

lim

s

→

0

+

⁡

ζ

⁢

(

s

)

=

lim

s

→

0

+

⁡

2

s

⁢

π

s

-

1

⁢

sin

⁡

π

⁢

s

2

⁢

Γ

⁢

(

1

-

s

)

⁢

ζ

⁢

(

1

-

s

)

=

lim

s

→

0

+

⁡

2

s

⁢

π

s

-

1

⁢

(

∑

n

=

0

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

(

π

⁢

s

2

)

2

⁢

n

+

1

)

⁢

Γ

⁢

(

1

-

s

)

⁢

(

1

(

1

-

s

)

-

1

+

1

-

(

1

-

s

)

⁢

∫

1

∞

x

-

[

x

]

x

(

1

-

s

)

+

1

⁢

𝑑

x

)

=

lim

s

→

0

+

⁡

2

s

⁢

π

s

-

1

⁢

(

π

⁢

s

2

)

⁢

(

∑

n

=

0

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

(

π

⁢

s

2

)

2

⁢

n

)

⁢

Γ

⁢

(

1

-

s

)

⁢

(

1

-

s

+

1

-

(

1

-

s

)

⁢

∫

1

∞

x

-

[

x

]

x

2

-

s

⁢

𝑑

x

)

=

lim

s

→

0

+

⁡

2

s

⁢

π

s

-

1

⁢

(

π

2

)

⁢

(

1

+

∑

n

=

1

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

(

π

⁢

s

2

)

2

⁢

n

)

⁢

Γ

⁢

(

1

-

s

)

⁢

s

⁢

(

-

1

s

+

1

-

(

1

-

s

)

⁢

∫

1

∞

x

-

[

x

]

x

2

-

s

⁢

𝑑

x

)

=

lim

s

→

0

+

⁡

2

s

-

1

⁢

π

s

⁢

(

1

+

∑

n

=

1

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

(

π

⁢

s

2

)

2

⁢

n

)

⁢

Γ

⁢

(

1

-

s

)

⁢

(

-

1

+

s

-

s

⁢

(

1

-

s

)

⁢

∫

1

∞

x

-

[

x

]

x

2

-

s

⁢

𝑑

x

)

=

(

lim

s

→

0

+

⁡

2

s

-

1

⁢

π

s

⁢

Γ

⁢

(

1

-

s

)

⁢

(

-

1

+

s

-

s

⁢

(

1

-

s

)

⁢

∫

1

∞

x

-

[

x

]

x

2

-

s

⁢

𝑑

x

)

)

⁢

(

lim

s

→

0

+

⁡

1

+

∑

n

=

1

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

(

π

⁢

s

2

)

2

⁢

n

)

=

(

2

0

-

1

⁢

π

0

⁢

Γ

⁢

(

1

-

0

)

⁢

(

-

1

+

0

-

0

⁢

(

1

-

0

)

⁢

∫

1

∞

x

-

[

x

]

x

2

-

0

⁢

𝑑

x

)

)

⁢

(

1

+

∑

n

=

1

∞

(

-

1

)

n

(

2

⁢

n

+

1

)

!

⁢

(

π

⋅

0

2

)

2

⁢

n

)

=

(

1

2

⋅

1

⋅

Γ

⁢

(

1

)

⋅

(

-

1

+

0

-

0

)

)

⁢

(

1

+

∑

n

=

1

∞

0

)

=

-

1

2

.

∎

Title

value of the Riemann zeta function at

s

=

0

Canonical name

ValueOfTheRiemannZetaFunctionAtS0

Date of creation

2013-03-22 16:07:17

Last modified on

2013-03-22 16:07:17

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

20

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11M06

Related topic

CriticalStrip

Related topic

FormulaeForZetaInTheCriticalStrip

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