aarjaneiro/planetmath huggingface dataset:
Factorization of the Dedekind zeta function of an abelian number field

Factorization of the Dedekind zeta function of an abelian number field

The

Dedekind zeta function

of an

abelian number field

factors as a product of

Dirichlet L-functions

as follows. Let

K

be an abelian number field, i.e.

K

/

ℚ

is Galois and

Gal

⁡

(

K

/

ℚ

)

is abelian. Then, by the

Kronecker-Weber theorem

,
there is an

integer

n

(which we choose to be minimal) such that

K

⊆

ℚ

⁢

(

ζ

n

)

where

ζ

n

is a primitive

n

th

root of unity

. Let

G

=

Gal

⁡

(

ℚ

⁢

(

ζ

n

)

/

ℚ

)

≅

(

ℤ

/

n

⁢

ℤ

)

×

and let

χ

:

G

→

ℂ

×

be a

Dirichlet character

. Then the kernel of

χ

determines a

fixed
field

of

ℚ

⁢

(

ζ

n

)

. Further, for any field

K

as before,
there exists a group

X

of Dirichlet characters of

G

such that

K

is equal to the

intersection

of the fixed fields by the
kernels of all

χ

∈

X

. The order of

X

is

[

K

:

ℚ

]

and

X

≅

Gal

⁡

(

K

/

ℚ

)

.

Theorem

(

[

1

]

, Thm. 4.3).

Let

K

be an abelian number field and let

X

be
the associated group of Dirichlet characters. The Dedekind zeta
function of

K

factors as follows:

ζ

K

⁢

(

s

)

=

∏

χ

∈

X

L

⁢

(

s

,

χ

)

.

Notice that for the trivial character

χ

0

one has

L

⁢

(

s

,

χ

0

)

=

ζ

⁢

(

s

)

, the

Riemann zeta function

, which has a

simple pole

at

s

=

1

with

residue

1

. Thus, for an arbitrary
abelian number field

K

:

ζ

K

⁢

(

s

)

=

∏

χ

∈

X

L

⁢

(

s

,

χ

)

=

ζ

⁢

(

s

)

⋅

∏

χ

0

≠

χ

∈

X

L

⁢

(

s

,

χ

)

where the last product is taken over all
non-trivial

characters

χ

∈

X

.

References

1

L. C. Washington,

Introduction to

Cyclotomic Fields

, Springer-Verlag, New York.

Title

Factorization of the Dedekind zeta function of an abelian number field

Canonical name

FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField

Date of creation

2013-03-22 16:01:21

Last modified on

2013-03-22 16:01:21

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11M06

Classification

msc 11R42

Related topic

ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers

Generated on Fri Feb 9 15:37:17 2018 by

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functional equation for the Riemann Xi function

functional equation for the Riemann Xi function

The Riemann Xi Function satisfies the following functional equation:

Ξ

⁢

(

s

)

=

Ξ

⁢

(

1

-

s

)

This equation directly implies the

Riemann Zeta function

’s functional equation.

This equation plays an important role in the theory of the Riemann Zeta function. It allows one to analytically continue the Zeta and the Xi

functions

to the whole

complex plane

. The definition of the

Zeta function

as a series is only valid when

ℜ

⁡

(

s

)

>

1

. By using this equation, one can express the values of these two functions when

ℜ

⁡

(

s

)

<

1

in terms of the values when

ℜ

⁡

(

s

)

>

1

. As an illustration of its importance, one can cite the fact that there are no zeros of the Zeta function with

real part

greater than 1, so without this functional equation the study of the Zeta function would be very limited.

Title

functional equation for the Riemann Xi function

Canonical name

FunctionalEquationForTheRiemannXiFunction

Date of creation

2013-03-22 13:24:15

Last modified on

2013-03-22 13:24:15

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 11M06

Generated on Fri Feb 9 15:37:19 2018 by

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functional equation for the theta function

functional equation for the theta function

The lemma used in the

derivation

of the

functional equation

for the Riemann Xi function. This functional equation is not as remarkable as the one for the Xi

function

, because it does not actually extend the domain of the function.

θ

⁢

(

1

x

)

=

x

⁢

θ

⁢

(

x

)

The proof relies on the

Cauchy integral formula

and the

Poisson summation formula

.

Title

functional equation for the

theta function

Canonical name

FunctionalEquationForTheThetaFunction

Date of creation

2013-03-22 13:24:03

Last modified on

2013-03-22 13:24:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 11M06

Generated on Fri Feb 9 15:37:20 2018 by

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generalized Riemann hypothesis

generalized Riemann hypothesis

This

generalization

of the

Riemann hypothesis

to arbitrary

Dedekind zeta functions

that for any

number field

K

, the only zeroes

s

of the Dedekind zeta function

ζ

K

⁢

(

s

)

that lie in the strip

0

≤

ℜ

⁡

s

≤

1

ℜ

⁡

s

=

1

2

.

Title

generalized Riemann hypothesis

Canonical name

GeneralizedRiemannHypothesis

Date of creation

2013-03-22 13:54:31

Last modified on

2013-03-22 13:54:31

Owner

mathcam (2727)

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mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11M06

Generated on Fri Feb 9 15:37:22 2018 by

LaTeXML
Lindelöf hypothesis

Lindelöf hypothesis

Ernst Lindelöf (

http://planetmath.org/ernstlindelof

) presented in 1908 a

conjecture

known as

Lindelöf

hypothesis

which concerns
the of the

Riemann zeta function

on the
“

critical line

”

x

=

1

2

.  Up to now (2015), this
hypothesis has not been proved.  It is weaker than the

Riemann hypothesis

such that this latter implies it but not conversely.

Lindelöf hypothesis.

ζ

⁢

(

1

2

+

i

⁢

t

)

=

O

⁢

(

t

ε

)

for every

ε

>

0

when

t

→

∞

.

Here

O

is the Landau
big ordo (

http://planetmath.org/formaldefinitionoflandaunotation

) notation.

References

1

Ernst Lindelöf

: “Quelques remarques sur
la croissance de la fonction

ζ

⁢

(

s

)

”. –

Bull. Sci. Math.

32

(1908).

Title

Lindelöf hypothesis

Canonical

name

LindelofHypothesis

Date of creation

2015-08-22 13:08:30

Last modified on

2015-08-22 13:08:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Conjecture

Classification

msc 11M06

Generated on Fri Feb 9 15:37:23 2018 by

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proof of functional equation for the theta function

proof of functional equation for the theta function

All sums are over all integers unless otherwise specified. Thus the

theta function

is

θ

⁢

(

x

)

=

∑

n

e

-

π

⁢

n

2

⁢

x

.

Using the
Jacobi’s

identity

for

ϑ

functions

(

http://planetmath.org/JacobisIdentityForVarthetaFunctions

) with

z

=

0

and

τ

=

i

/

x

,
so that

-

1

/

τ

=

i

⁢

x

gives

θ

3

(

0

∣

i

x

)

=

(

1

/

x

)

1

/

2

θ

3

(

0

∣

i

/

x

)

.

Using the definition of

θ

3

(

http://planetmath.org/JacobiVarthetaFunctions

) we have that the left hand side is

∑

n

e

-

π

⁢

x

⁢

n

2

=

θ

⁢

(

x

)

while the right hand side is

(

1

/

x

)

1

/

2

⁢

∑

n

e

i

⁢

π

⁢

(

i

/

x

)

⁢

n

2

which is

(

1

/

x

)

1

/

2

⁢

∑

n

e

-

π

⁢

n

2

/

x

=

1

x

⁢

θ

⁢

(

1

/

x

)

so the identity is established.

The identity is attributed to Poisson by Jacobi

[

1

]

. Jacobi writes:
M. Poisson, dans ses savantes recheches sur les intégrales définies, a
fait connaître plusieurs propriétés de la fonction

Θ

⁢

(

x

)

. Les méthodes
délicates, propres à cet illustre géomètre, trouvent une belle vérification
dans la théorie des fonctions elliptiques. Par exemple, M. Poisson démontre
dans dix-neuvième cahier du Journal de l’école polytechnique la formule remarquable:

1

x

=

1

+

2

⁢

e

-

π

⁢

x

+

2

⁢

e

-

4

⁢

π

⁢

x

+

2

⁢

e

-

9

⁢

π

⁢

x

+

2

⁢

e

-

16

⁢

π

⁢

x

+

⋯

1

+

2

⁢

e

-

π

x

+

2

⁢

e

-

4

⁢

π

x

+

2

⁢

e

-

9

⁢

π

x

+

2

⁢

e

-

16

⁢

π

x

+

⋯

References

1

M.C.G.J Jacobi,

Notices sur Les Fonctions Elliptiques

,
in Jacobi’s Gesammelte Werke, Band 1, Berlin, 1881, page 260.

Title

proof of

functional equation

for the theta function

Canonical name

ProofOfFunctionalEquationForTheThetaFunction

Date of creation

2013-03-22 13:27:27

Last modified on

2013-03-22 13:27:27

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

14

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 11M06

Generated on Fri Feb 9 15:37:26 2018 by

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properties of Riemann xi function

properties of Riemann xi function

The Riemann xi function, defined by

ξ

⁢

(

s

)

:=

s

2

⁢

(

s

-

1

)

⁢

π

-

s

2

⁢

Γ

⁢

(

s

2

)

⁢

ζ

⁢

(

s

)

,

is an

entire function

having as zeros the nonreal zeros of the

Riemann zeta function

ζ

and only them.

The

modulus

of the xi

function

is strictly increasing along every horizontal half-line lying
in any open

right half-plane

that contains no xi zeros.  As well, the modulus decreases strictly along
every horizontal half-line in any zero-free, open

left half-plane

.

Taking into account the functional equation

ξ

⁢

(

1

-

s

)

=

ξ

⁢

(

s

)

it follows the reformulation of the

Riemann hypothesis

:

Theorem.

The following three statements are

equivalent

.

(i). If

t

is any fixed

real number

, then

|

ξ

⁢

(

σ

+

i

⁢

t

)

|

is increasing for

1

2

<

σ

<

∞

.

(ii). If

t

is any fixed real number, then

|

ξ

⁢

(

σ

+

i

⁢

t

)

|

is decreasing for

-

∞

<

σ

<

1

2

.

(iii). The Riemann hypothesis is true.

References

1

Jonathan Sondow & Christian Dumitrescu

: A monotonicity property Riemann’s xi function and a reformulation of the Riemann Hypothesis. –

Periodica

Mathematica

Hungarica

60

(2010) 37–40. Also available

http://arxiv.org/ftp/arxiv/papers/1005/1005.1104.pdf

here.

Title

properties of Riemann xi function

Canonical name

PropertiesOfRiemannXiFunction

Date of creation

2013-03-22 19:35:25

Last modified on

2013-03-22 19:35:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Result

Classification

msc 11M06

Related topic

RobinsTheorem

Related topic

ExtraordinaryNumber

Generated on Fri Feb 9 15:37:28 2018 by

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Riemann Ξ function

Riemann

Ξ

function

The

Riemann Xi function

Ξ

⁢

(

s

)

=

π

-

1

2

⁢

s

⁢

Γ

⁢

(

1

2

⁢

s

)

⁢

ζ

⁢

(

s

)

,

(where

Γ

⁢

(

s

)

is Euler’s

Gamma function

and

ζ

⁢

(

s

)

is the

Riemann zeta function

), is the key to the functional equation for the Riemann zeta function.

Riemann himself used the notation of a lower case xi (

ξ

). The famous

Riemann hypothesis

is equivalent to the assertion that all the zeros of

ξ

are real, in fact Riemann himself presented his original

hypothesis

in terms of that

function

.

Riemann’s lower case xi is defined as

ξ

⁢

(

s

)

=

1

2

⁢

s

⁢

(

s

-

1

)

⁢

Ξ

⁢

(

s

)

.

Title

Riemann

Ξ

function

Canonical

name

RiemannXiFunction

Date of creation

2013-03-22 13:24:06

Last modified on

2013-03-22 13:24:06

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

11

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11M06

Generated on Fri Feb 9 15:37:30 2018 by

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Riemann θ-function

Riemann

θ

-function

The

Riemann theta function

is a number-theoretic function
which is only really used in
the

derivation

of the functional equation for the Riemann xi function.

The Riemann theta function is defined as:

θ

⁢

(

x

)

=

2

⁢

ω

⁢

(

x

)

+

1

,

where

ω

is the Riemann omega function.

The domain (

http://planetmath.org/

Function

) of the Riemann theta function is

x

>

0

.

To give an explicit form for the

theta function

, note that

ω

⁢

(

x

)

=

∑

n

=

1

∞

e

-

n

2

⁢

π

⁢

x

=

∑

n

=

-

1

-

∞

e

-

(

-

n

)

2

⁢

π

⁢

x

=

∑

n

=

-

1

-

∞

e

-

n

2

⁢

π

⁢

x

and so

2

⁢

ω

⁢

(

x

)

+

1

=

∑

n

=

-

1

-

∞

e

-

n

2

⁢

π

⁢

x

+

ω

⁢

(

x

)

+

1

=

∑

n

=

-

1

-

∞

e

-

n

2

⁢

π

⁢

x

+

∑

n

=

1

∞

e

-

n

2

⁢

π

⁢

x

+

e

-

0

2

⁢

π

⁢

x

=

∑

n

=

-

∞

∞

e

-

n

2

⁢

π

⁢

x

.

Thus we have

θ

⁢

(

x

)

=

∑

n

=

-

∞

∞

e

-

n

2

⁢

π

⁢

x

.

Riemann showed that the theta function satisfied a functional equation,
which was the key step
in the proof of the analytic continuation for the Riemann xi function.
This has direct

consequences

for the

Riemann zeta function

.

Title

Riemann

θ

-function

Canonical name

Riemannthetafunction

Date of creation

2013-03-22 13:23:58

Last modified on

2013-03-22 13:23:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

13

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11M06

Synonym

Riemann theta-function

Synonym

Riemann theta function

Related topic

LandsbergSchaarRelation

Generated on Fri Feb 9 15:37:37 2018 by

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Riemann ϖ function

Riemann

ϖ

function

The Riemann

ϖ

function

is used in the proof of the

analytic continuation

for the Riemann Xi function to the whole

complex plane

. It is defined as:

ϖ

⁢

(

x

)

=

∑

n

=

1

∞

e

-

n

2

⁢

π

⁢

x

This function is a special case of a Jacobi

ϑ

function (

http://planetmath.org/JacobiVarthetaFunctions

):

ϖ

(

x

)

=

ϑ

3

(

0

|

i

x

)

As such the

ϖ

function satisfies a

functional equation

, which a special case of Jacobi’s Identity for the

ϑ

function (

http://planetmath.org/JacobisIdentityForVarthetaFunctions

).

Title

Riemann

ϖ

function

Canonical name

RiemannvarpiFunction

Date of creation

2013-03-22 13:24:12

Last modified on

2013-03-22 13:24:12

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 11M06

Generated on Fri Feb 9 15:37:39 2018 by

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Siegel-Klingen Theorem

Siegel-Klingen Theorem

Theorem

(Siegel-Klingen Theorem,

[

1

]

,

[

2

]

).

Let

K

be a totally real number field and let

ζ

⁢

(

s

,

K

)

be the

Dedekind zeta function

of

K

. If

n

≥

1

is an

integer

then

ζ

⁢

(

-

n

,

K

)

is a

rational number

(i.e.

ζ

⁢

(

-

n

,

K

)

∈

Q

).

References

1

Klingen, Helmut,

Über die Werte der Dedekindschen Zetafunktion

. (German) Math. Ann. 145 1961/1962 265–272.

2

Siegel, Carl Ludwig,

Über die analytische Theorie der quadratischen Formen. III

. (German) Ann. of Math. (2) 38 (1937), no. 1, 212–291.

Title

Siegel-Klingen Theorem

Canonical name

SiegelKlingenTheorem

Date of creation

2013-03-22 16:01:19

Last modified on

2013-03-22 16:01:19

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

4

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11M06

Classification

msc 11R42

Generated on Fri Feb 9 15:37:40 2018 by

LaTeXML
value of Riemann zeta function at s=4

value of Riemann zeta function at

s

=

4

By applying Parseval’s

identity

(Lyapunov equation (

http://planetmath.org/PersevalEquality

)) to the

Fourier series

a

0

2

+

(

a

1

⁢

cos

⁡

x

+

b

1

⁢

sin

⁡

x

)

+

(

a

2

⁢

cos

⁡

2

⁢

x

+

b

2

⁢

sin

⁡

2

⁢

x

)

+

…

of

x

2

on the

interval

[

-

π

,

π

]

,  one may derive the value of

Riemann zeta function

at

s

=

4

.

Let us first find the needed Fourier coefficients

a

n

and

b

n

.  Since

x

2

defines an

even function

, we have

b

n

=

0

∀

n

=

1

,

2

,

3

,

…

.

Then

a

0

=

1

π

⁢

∫

-

π

π

x

2

⁢

𝑑

x

=

2

π

⁢

∫

0

π

x

2

⁢

𝑑

x

=

2

⁢

π

2

3

.

For other

coefficients

a

n

, we must perform twice

integrations by parts

:

a

n

=

1

π

⁢

∫

-

π

π

x

2

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

=

2

π

⁢

∫

0

π

x

2

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

=

2

π

⁢

(

/

0

π

⁡

x

2

⋅

sin

⁡

n

⁢

x

n

-

∫

0

π

2

⁢

x

⋅

sin

⁡

n

⁢

x

n

⁢

𝑑

x

)

=

-

4

n

⁢

π

⁢

∫

o

π

x

⁢

sin

⁡

n

⁢

x

⁢

d

⁢

x

=

-

4

n

⁢

π

⁢

(

/

0

π

⁡

x

⋅

-

cos

⁡

n

⁢

x

n

-

∫

0

π

1

⋅

-

cos

⁡

n

⁢

x

n

⁢

𝑑

x

)

=

-

4

n

⁢

π

⁢

/

0

π

⁡

(

-

x

⁢

cos

⁡

n

⁢

x

n

-

sin

⁡

n

⁢

x

n

2

)

=

4

⁢

cos

⁡

n

⁢

π

n

2

=

4

⁢

(

-

1

)

n

n

2

∀

n

=

1

,

2

,

3

,

…

Thus

x

2

=

π

2

3

+

∑

n

=

1

∞

4

⁢

(

-

1

)

n

n

2

⁢

cos

⁡

n

⁢

x

for

-

π

≦

x

≦

π

.

The

left hand side

of Parseval’s identity

1

2

⁢

π

⁢

∫

-

π

π

(

f

⁢

(

x

)

)

2

⁢

𝑑

x

=

a

0

2

4

+

1

2

⁢

∑

n

=

1

∞

(

a

n

2

+

b

n

2

)

reads now

1

π

⁢

∫

0

π

(

x

2

)

2

⁢

𝑑

x

=

1

π

⁢

/

0

π

⁡

x

5

5

=

π

4

5

and its right hand side

1

4

⁢

(

2

⁢

π

2

3

)

2

+

1

2

⁢

∑

n

=

1

∞

(

4

n

2

)

2

=

π

4

9

+

8

⁢

∑

n

=

1

∞

1

n

4

=

π

4

9

+

8

⁢

ζ

⁢

(

4

)

.

Accordingly, we obtain the result

ζ

⁢

(

4

)

=

1

+

1

2

4

+

1

3

4

+

…

=

π

4

90

.

(1)

Title

value of Riemann zeta function at

s

=

4

Canonical name

ValueOfRiemannZetaFunctionAtS4

Date of creation

2013-03-22 18:22:06

Last modified on

2013-03-22 18:22:06

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Example

Classification

msc 11M06

Related topic

SubstitutionNotation

Related topic

CosineAtMultiplesOfStraightAngle

Related topic

ValueOfTheRiemannZetaFunctionAtS2

Generated on Fri Feb 9 15:37:43 2018 by

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extraordinary number

extraordinary number

Define the function

G

for

integers

n

>

1

by

G

⁢

(

n

)

:=

σ

⁢

(

n

)

n

⁢

ln

⁡

(

ln

⁡

n

)

,

where

σ

⁢

(

n

)

is the sum of the

positive

divisors

of

n

.  A

positive integer

N

is said to be an

extraordinary number

if it is

composite

and

G

⁢

(

N

)

≥

max

⁡

{

G

⁢

(

N

/

p

)

,

G

⁢

(

a

⁢

N

)

}

for any

prime factor

p

of

N

and any

multiple

a

⁢

N

of

N

.

It has been proved in [1] that the

Riemann Hypothesis

is true iff 4 is the only extraordinary number.  The proof is based on Gronwall’s theorem and Robin’s theorem.

References

1

Geoffrey Caveney, Jean-Louis Nicolas, Jonathan Sondow

:  Robin’s theorem, primes, and a new elementary reformulation of the Riemann Hypothesis.

-

Integers

11

(2011) article A33;  available directly at

http://arxiv.org/pdf/1110.5078.pdf

arXiv.

Title

extraordinary number

Canonical name

ExtraordinaryNumber

Date of creation

2013-03-22 19:33:41

Last modified on

2013-03-22 19:33:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Definition

Classification

msc 11M26

Classification

msc 11A25

Related topic

PropertiesOfXiFunction

Generated on Fri Feb 9 15:37:49 2018 by

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generalized zeta function

generalized zeta function

The

generalized zeta function

is defined for

σ

⩾

1

+

δ

by

ζ

⁢

(

s

,

a

)

=

∑

n

=

0

∞

1

(

a

+

n

)

s

,

where

a

is a constant. Clearly,

ζ

⁢

(

s

,

1

)

=

ζ

⁢

(

s

)

, where

ζ

⁢

(

s

)

is the

Riemann zeta function

.

References

1

Hurwitz,

Zeitschritf für Math. und Phys

. xxvii. (1882), pp. 86-101.

Title

generalized zeta function

Canonical name

GeneralizedZetaFunction

Date of creation

2013-03-22 16:11:35

Last modified on

2013-03-22 16:11:35

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11M35

Generated on Fri Feb 9 15:37:51 2018 by

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Dirichlet eta function

Dirichlet eta function

For

s

∈

ℂ

, the

Dirichlet eta function

is defined as

η

⁢

(

s

)

:=

∑

n

=

1

∞

(

-

1

)

n

-

1

n

s

.

(1)

Let

s

=

σ

+

i

⁢

t

. For

s

a positive real number the series

converges

by the

alternating series test

, by the second listed in the entry on

Dirichlet series

it converges for all

s

with

σ

>

0

.

It can be shown that

η

⁢

(

s

)

=

(

1

-

2

1

-

s

)

⁢

ζ

⁢

(

s

)

, where

ζ

⁢

(

s

)

is the

Riemann zeta function

. The pole of

ζ

⁢

(

s

)

at

s

=

1

is cancelled by the zero
of

1

-

2

1

-

s

.

Title

Dirichlet eta function

Canonical name

DirichletEtaFunction

Date of creation

2013-03-22 14:31:28

Last modified on

2013-03-22 14:31:28

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11M41

Related topic

ZerosOfDirichletEtaFunction

Generated on Fri Feb 9 15:37:52 2018 by

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proof that η⁢(1)=l⁢n2

proof that

η

⁢

(

1

)

=

l

⁢

n

2

Theorem 1

.

η

⁢

(

1

)

=

ln

⁡

2

, where

η

is the

Dirichlet eta function

.

Proof.

By

definition

,

η

⁢

(

1

)

=

∑

n

=

1

∞

(

-

1

)

n

+

1

n

=

-

∑

n

=

1

∞

(

-

1

)

n

n

.

Applying Abel’s Limit Theorem,

η

⁢

(

1

)

=

-

lim

r

→

1

-

⁡

∑

n

=

1

∞

(

-

r

)

n

n

=

lim

r

→

1

-

⁡

ln

⁡

(

1

+

r

)

=

ln

⁡

2

Title

proof that

η

⁢

(

1

)

=

l

⁢

n

2

Canonical

name

ProofThateta1ln2

Date of creation

2013-03-22 17:57:20

Last modified on

2013-03-22 17:57:20

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11M41

Generated on Fri Feb 9 15:37:55 2018 by

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value of Dirichlet eta function at s=2

value of Dirichlet eta function at

s

=

2

The value

η

(

2

)

=

1

-

1

2

2

+

1

3

2

-

1

4

2

+

-

…

of the

Dirichlet eta function

can be found by using the

Fourier cosine series

of the

function

x

↦

x

-

x

2

on the

interval

[

0

,

1

]

:

x

-

x

2

=

1

6

-

1

π

2

⁢

∑

n

=

1

∞

cos

⁡

2

⁢

n

⁢

π

⁢

x

n

2

for

⁢

0

≦

x

≦

1

(1)

Substituting

x

:=

1

2

to the equation (1) yields

1

4

=

1

6

-

1

π

2

⁢

∑

n

=

1

∞

cos

⁡

n

⁢

π

n

2

=

1

6

+

1

π

2

⁢

∑

n

=

1

∞

(

-

1

)

n

+

1

n

2

,

which we can solve to the form

η

⁢

(

2

)

=

∑

n

=

1

∞

(

-

1

)

n

+

1

n

2

=

π

2

12

.

(2)

This result could be obtained very simply by using the functional equation connecting Dirichlet eta function to

Riemann zeta function

.

Combining the equation (2) with the result concerning the Riemann zeta function at 2 (

http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2

) shows that

1

+

1

3

2

+

1

5

2

+

1

7

2

+

…

=

π

2

8

.

(3)

Title

value of Dirichlet eta function at

s

=

2

Canonical name

ValueOfDirichletEtaFunctionAtS2

Date of creation

2013-03-22 18:22:09

Last modified on

2013-03-22 18:22:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Result

Classification

msc 11M41

Related topic

CosineAtMultiplesOfStraightAngle

Related topic

ValueOfTheRiemannZetaFunctionAtS2

Generated on Fri Feb 9 15:37:57 2018 by

LaTeXML
η⁢(1)=l⁢n2

η

⁢

(

1

)

=

l

⁢

n

2

Since

ζ

⁢

(

1

)

=

∞

,

η

⁢

(

1

)

cannot be computed as indicated in the

Dirichlet eta function

entry.

η

⁢

(

1

)

=

ln

⁡

2

which is the alternate

harmonic series of order

1.

Title

η

⁢

(

1

)

=

l

⁢

n

2

Canonical name

eta1ln2

Date of creation

2013-03-22 16:10:30

Last modified on

2013-03-22 16:10:30

Owner

dextercioby (12657)

Last modified by

dextercioby (12657)

Numerical id

5

Author

dextercioby (12657)

Entry type

Example

Classification

msc 11M41

Generated on Fri Feb 9 15:37:59 2018 by

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