aarjaneiro/planetmath huggingface dataset:
uniformly distributed

uniformly distributed

Let

{

u

n

}

be a

sequence

of

real numbers

. For

0

≤

α

<

β

≤

1

put

Z

(

N

,

α

,

β

)

=

card

{

n

∈

[

1

.

.

N

]

:

α

≤

(

u

n

mod

1

)

<

β

}

.

The sequence

{

u

n

}

is

uniformly distributed

modulo

1

if

lim

N

→

∞

⁡

1

N

⁢

Z

⁢

(

N

,

α

,

β

)

=

β

-

α

for all

0

≤

α

<

β

≤

1

. In other words a sequence is
uniformly distributed modulo

1

if each

subinterval

of

[

0

,

1

]

gets its “fair share” of

fractional parts

of

{

u

n

}

.

More generally, a sequence

{

u

n

}

of points in a finite measure
space

(

X

,

𝒜

,

μ

)

is uniformly distributed with respect
to a family of sets

ℱ

⊂

𝒜

if

lim

N

→

∞

⁡

card

{

n

∈

[

1

.

.

N

]

:

u

n

∈

S

}

N

=

μ

⁢

(

S

)

μ

⁢

(

X

)

for every

⁢

S

∈

ℱ

.

References

1

William Chen.

Lectures on irregularities of point distribution.

Available at

http://www.maths.mq.edu.au/ wchen/ln.html

http://www.maths.mq.edu.au/ wchen/ln.html, 2000.

2

Hugh L. Montgomery.

Ten Lectures on the Interface Between

Analytic Number Theory

and

Harmonic Analysis

, volume 84 of

Regional Conference Series in
Mathematics

.

AMS, 1994.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0814.11001

Zbl 0814.11001.

Title

uniformly distributed

Canonical

name

UniformlyDistributed

Date of creation

2013-03-22 14:17:29

Last modified on

2013-03-22 14:17:29

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Definition

Classification

msc 11K06

Classification

msc 11K38

Synonym

equidistributed

Related topic

WeylsCriterion

Generated on Fri Feb 9 15:36:25 2018 by

LaTeXML
Weyl’s criterion

Weyl’s criterion

Let

{

u

n

}

be a

sequence

of real numbers. Then

{

u

n

}

is

uniformly distributed

modulo

1

if and only if

lim

N

→

∞

⁡

1

N

⁢

∑

n

=

1

N

e

⁢

(

k

⁢

u

n

)

=

0

for every nonzero

integer

k

, where

e

⁢

(

x

)

=

exp

⁡

(

2

⁢

π

⁢

i

⁢

x

)

.

Weyl’s criterion reduces the problem of

uniform distribution

of
sequences to the problem of estimating certain exponential sums.
Whereas the problem of estimating a family of exponential sums
might seem harder at first, the exponential map has the

multiplicative

property

which often makes the problem easier.

Example:

If

x

is

irrational

, then the sequence

{

n

⁢

x

}

is uniformly distributed modulo

1

. Proof:

|

∑

n

=

1

N

e

⁢

(

k

⁢

n

⁢

x

)

|

=

|

e

⁢

(

k

⁢

(

N

+

1

)

⁢

x

)

-

e

⁢

(

k

⁢

x

)

e

⁢

(

k

⁢

x

)

-

1

|

≤

2

|

e

⁢

(

k

⁢

x

)

-

1

|

=

O

k

⁢

(

1

)

because the irrationality of

x

implies

e

⁢

(

k

⁢

x

)

≠

1

.

References

1

Ã?. Ã?. ÃÅ¡ÃÂ°Ãâ¬ÃÂ°Ãâ ÃÆÃÂ±ÃÂ°.

ÃÅ¾Ã?ÃÂ½ÃÂ¾ÃÂ²Ãâ¹ ÃÂ°ÃÂ½ÃÂ°ÃÂ»ÃÂ¸ÃâÃÂ¸Ãâ¡ÃÂµÃ?ÃÂºÃÂ¾ÃÂ¹ ÃâÃÂµÃÂ¾Ãâ¬ÃÂ¸ÃÂ¸ Ãâ¡ÃÂ¸Ã?ÃÂµÃÂ»

.

Ã?ÃÂ°ÃÆÃÂºÃÂ°, 1983.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0428.10019

Zbl 0428.10019.

For English

translation

see

[

2

]

.

2

A. A. Karatsuba.

Basic

analytic number theory

.

Springer-Verlag, 1993.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0767.11001

Zbl 0767.11001.

3

Hugh L. Montgomery.

Ten Lectures on the Interface Between Analytic Number Theory and

Harmonic Analysis

, volume 84 of

Regional Conference Series in
Mathematics

.

AMS, 1994.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0814.11001

Zbl 0814.11001.

Title

Weyl’s criterion

Canonical

name

WeylsCriterion

Date of creation

2013-03-22 14:17:31

Last modified on

2013-03-22 14:17:31

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11K06

Classification

msc 11K38

Classification

msc 11L03

Related topic

UniformlyDistributed

Generated on Fri Feb 9 15:36:31 2018 by

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random numbers

random numbers

Random numbers

are particular occurrences of

random variables

. They are necessary for

Monte Carlo

calculations as well as many other computerized processes. There are three kinds of random numbers:

•

truly random numbers

•

pseudorandom numbers

•

quasirandom numbers

Title

random numbers

Canonical name

RandomNumbers

Date of creation

2013-03-22 12:07:18

Last modified on

2013-03-22 12:07:18

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

5

Author

akrowne (2)

Entry type

Definition

Classification

msc 11K45

Classification

msc 65C10

Related topic

TrulyRandomNumbers

Related topic

PseudorandomNumbers

Related topic

QuasiRandomNumbers

Generated on Fri Feb 9 15:36:38 2018 by

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Ramanujan sum

Ramanujan sum

For

positive integers

s

and

n

, the

complex number

c

s

⁢

(

n

)

=

∑

0

≤

k

<

n

(

k

,

n

)

=

1

e

2

⁢

π

⁢

i

⁢

k

⁢

s

/

n

is referred to as a

Ramanujan sum

, or a

Ramanujan trigonometric sum

.
Since

e

2

⁢

π

⁢

i

=

1

, an

equivalent

definition is

c

s

⁢

(

n

)

=

∑

k

∈

r

⁢

(

n

)

e

2

⁢

π

⁢

i

⁢

k

⁢

s

/

n

where

r

⁢

(

n

)

is some

reduced residue system

mod

n

, meaning any
subset of

ℤ

containing exactly one element of each

invertible

residue class

mod

n

.

Using a

symmetry

argument

about

roots of unity

, one can show

∑

d

|

s

c

s

⁢

(

n

)

=

{

s

if

s

|

n

0

otherwise.

Applying Möbius

inversion

, we get

c

s

⁢

(

n

)

=

∑

d

|

n

d

|

s

μ

⁢

(

n

/

d

)

⁢

d

=

∑

d

|

(

n

,

s

)

μ

⁢

(

n

/

d

)

⁢

d

which shows that

c

s

⁢

(

n

)

is a

real number

, and indeed an

integer

.
In particular

c

s

⁢

(

1

)

=

μ

⁢

(

s

)

. More generally,

c

s

⁢

t

⁢

(

m

⁢

n

)

=

c

s

⁢

(

m

)

⁢

c

t

⁢

(

n

)

⁢

if

⁢

(

m

,

t

)

=

(

n

,

s

)

=

1

.

Using the

Chinese remainder theorem

, it is not hard to show that
for any fixed

n

, the

function

s

↦

c

s

⁢

(

n

)

is

multiplicative

:

c

s

⁢

(

n

)

⁢

c

t

⁢

(

n

)

=

c

s

⁢

t

⁢

(

n

)

⁢

if

⁢

(

s

,

t

)

=

1

.

If

m

is invertible mod

n

, then the

mapping

k

↦

k

⁢

m

is a

permutation

of the invertible residue classes mod

n

. Therefore

c

s

⁢

(

m

⁢

n

)

=

c

s

⁢

(

n

)

⁢

if

⁢

(

m

,

s

)

=

1

.

Remarks:

Trigonometric sums often make
convenient apparatus in

number theory

, since any
function on a quotient ring of

ℤ

defines
a periodic function on

ℤ

itself, and

conversely

. For
another example, see

Landsberg-Schaar relation

.

Some writers use different notation from ours, reversing the roles
of

s

and

n

in the expression

c

s

⁢

(

n

)

.

The name “Ramanujan sum” was introduced by Hardy.

Title

Ramanujan sum

Canonical name

RamanujanSum

Date of creation

2013-03-22 12:11:57

Last modified on

2013-03-22 12:11:57

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 11L03

Classification

msc 11T23

Related topic

RootOfUnity

Defines

Ramanujan trigonometric sum

Generated on Fri Feb 9 15:36:40 2018 by

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derivation of Gauss sum up to a sign

derivation of Gauss sum up to a sign

The

Gauss sum

can be easily evaluated up to a sign by squaring the original series

g

1

2

⁢

(

χ

)

=

∑

s

∈

ℤ

/

p

⁢

ℤ

(

s

p

)

⁢

e

2

⁢

π

⁢

i

⁢

s

/

p

⁢

∑

t

∈

ℤ

/

p

⁢

ℤ

(

t

p

)

⁢

e

2

⁢

π

⁢

i

⁢

t

/

p

=

∑

s

,

t

∈

ℤ

/

p

⁢

ℤ

(

s

⁢

t

p

)

⁢

e

2

⁢

π

⁢

i

⁢

(

s

+

t

)

/

p

and

summing

over a new variable

n

=

s

-

1

⁢

t

(

mod

p

)

=

∑

s

,

n

∈

ℤ

/

p

⁢

ℤ

(

n

p

)

⁢

e

2

⁢

π

⁢

i

⁢

(

s

+

n

⁢

s

)

=

∑

n

∈

ℤ

/

p

⁢

ℤ

(

n

p

)

⁢

∑

s

∈

ℤ

/

p

⁢

ℤ

e

2

⁢

π

⁢

i

⁢

s

⁢

(

n

+

1

)

=

∑

n

∈

ℤ

/

p

⁢

ℤ

(

n

p

)

(

q

[

n

≡

-

1

(

mod

p

)

]

-

1

)

=

p

⁢

(

-

1

p

)

-

∑

n

∈

ℤ

/

p

⁢

ℤ

(

n

p

)

=

p

⁢

(

-

1

p

)

=

{

p

,

if

⁢

p

≡

1

(

mod

4

)

,

-

p

,

if

⁢

p

≡

3

(

mod

4

)

.

References

1

Harold Davenport.

Multiplicative Number Theory

.

Markham Pub. Co., 1967.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0159.06303

Zbl 0159.06303.

Title

derivation of Gauss sum up to a sign

Canonical name

DerivationOfGaussSumUpToASign

Date of creation

2013-03-22 13:39:45

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2013-03-22 13:39:45

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Derivation

Classification

msc 11L05

Generated on Fri Feb 9 15:36:43 2018 by

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Gauss sum

Gauss sum

Let

p

be a prime. Let

χ

be any

multiplicative group

character

on

ℤ

/

p

⁢

ℤ

(that is, any

group homomorphism

of multiplicative groups

(

ℤ

/

p

⁢

ℤ

)

×

→

ℂ

×

). For any

a

∈

ℤ

/

p

⁢

ℤ

, the

complex number

g

a

⁢

(

χ

)

:=

∑

t

∈

ℤ

/

p

⁢

ℤ

χ

⁢

(

t

)

⁢

e

2

⁢

π

⁢

i

⁢

a

⁢

t

/

p

is called a

Gauss sum

on

ℤ

/

p

⁢

ℤ

associated to

χ

.

In general, the

equation

g

a

⁢

(

χ

)

=

χ

⁢

(

a

-

1

)

⁢

g

1

⁢

(

χ

)

(for nontrivial

a

and

χ

) reduces the computation of general Gauss sums to that of

g

1

⁢

(

χ

)

. The

absolute value

of

g

1

⁢

(

χ

)

is always

p

as long as

χ

is nontrivial, and if

χ

is a quadratic character (that is,

χ

⁢

(

t

)

is the

Legendre symbol

(

t

p

)

), then the value of the Gauss sum is known to be

g

1

⁢

(

χ

)

=

{

p

,

p

≡

1

(

mod

4

)

,

i

⁢

p

,

p

≡

3

(

mod

4

)

.

References

1

Kenneth Ireland & Michael Rosen,

A Classical Introduction to Modern

Number Theory

, Second Edition, Springer–Verlag, 1990.

Title

Gauss sum

Canonical name

GaussSum

Date of creation

2013-03-22 12:48:28

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2013-03-22 12:48:28

Owner

djao (24)

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Numerical id

7

Author

djao (24)

Entry type

Definition

Classification

msc 11L05

Related topic

KloostermanSum

Generated on Fri Feb 9 15:36:46 2018 by

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Kloosterman sum

Kloosterman sum

The

Kloosterman sum

is one of various trigonometric sums that are
useful in

number theory

and, more generally, in finite harmonic
analysis.
The original Kloosterman sum is

K

p

⁢

(

a

,

b

)

=

∑

x

∈

𝔽

p

*

exp

⁡

(

2

⁢

π

⁢

i

⁢

(

a

⁢

x

+

b

⁢

x

-

1

)

p

)

where

𝔽

p

is the field of prime order

p

.
Such sums have been generalized
in a few different ways since their introduction in 1926.
For instance, let

q

be a prime power,

𝔽

q

the field
of

q

elements,

χ

:

𝔽

q

*

→

ℂ

a

character

, and

ψ

:

𝔽

q

→

ℂ

a mapping such that

ψ

⁢

(

x

+

y

)

=

ψ

⁢

(

x

)

⁢

ψ

⁢

(

y

)

identically.
The sums

K

ψ

(

χ

|

a

,

b

)

=

∑

x

∈

𝔽

q

*

χ

(

x

)

ψ

(

a

x

+

b

x

-

1

)

are of

interest

, because they come up as

Fourier coefficients

of

modular forms

.

Kloosterman sums are finite analogs of the

K

-

Bessel
functions

of this kind:

K

s

⁢

(

a

)

=

1

2

⁢

∫

0

∞

x

s

-

1

⁢

exp

⁡

(

-

a

⁢

(

x

+

x

-

1

)

2

)

⁢

𝑑

x

where

ℜ

⁡

(

a

)

>

0

.

Title

Kloosterman sum

Canonical

name

KloostermanSum

Date of creation

2013-03-22 13:59:33

Last modified on

2013-03-22 13:59:33

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11L05

Classification

msc 43A25

Related topic

GaussSum

Generated on Fri Feb 9 15:36:58 2018 by

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Landsberg-Schaar relation

Landsberg-Schaar relation

The

Landsberg-Schaar relation

states that for any

positive integers

p

and

q

:

1

p

⁢

∑

n

=

0

p

-

1

exp

⁡

(

2

⁢

π

⁢

i

⁢

n

2

⁢

q

p

)

=

e

π

⁢

i

/

4

2

⁢

q

⁢

∑

n

=

0

2

⁢

q

-

1

exp

⁡

(

-

π

⁢

i

⁢

n

2

⁢

p

2

⁢

q

)

(1)

Although both sides of (1) are mere finite sums,
no one has yet found a proof which uses no

infinite

limiting process. One way to prove it is to put

τ

=

2

⁢

i

⁢

q

/

p

+

ϵ

, where

ϵ

>

0

, in
this

identity

due to Jacobi:

∑

n

=

-

∞

+

∞

e

-

π

⁢

n

2

⁢

τ

=

1

τ

⁢

∑

n

=

-

∞

+

∞

e

-

π

⁢

n

2

/

τ

(2)

and let

ϵ

→

0

. The details can be found here (

http://planetmath.org/ProofOfJacobisIdentityForVarthetaFunctions

). The identity (2) is a basic one in the theory of

theta functions

. It is sometimes called the functional equation for the

Riemann theta function

. See e.g. [2 VII.6.2].

If we just let

q

=

1

in the Landsberg-Schaar identity, it reduces to a

formula

for the quadratic Gauss sum mod

p

; notice that

p

need not be prime.

References:

[1] H. Dym and H.P. McKean.

Fourier Series

and

Integrals

. Academic Press, 1972.

[2] J.-P. Serre.

A Course in

Arithmetic

. Springer, 1970.

Title

Landsberg-Schaar relation

Canonical name

LandsbergSchaarRelation

Date of creation

2013-03-22 13:23:20

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2013-03-22 13:23:20

Owner

mathcam (2727)

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mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 11L05

Synonym

Schaar’s identity

Related topic

RiemannThetaFunction

Generated on Fri Feb 9 15:37:00 2018 by

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Pólya-Vinogradov inequality

Pólya-Vinogradov inequality

Theorem 1

.

For

m

,

n

∈

N

and

p

a

positive

odd

rational
prime

,

|

∑

t

=

m

m

+

n

(

t

p

)

|

<

p

⁢

ln

⁡

p

.

Proof.

Start with the following manipulations:

∑

t

=

m

m

+

n

(

t

p

)

=

1

p

⁢

∑

t

=

0

p

-

1

∑

x

=

m

m

+

n

∑

a

=

0

p

-

1

(

t

p

)

⁢

e

2

⁢

π

⁢

i

⁢

a

⁢

(

x

-

t

)

/

p

=

1

p

⁢

∑

a

=

1

p

-

1

∑

x

=

m

m

+

n

e

2

⁢

π

⁢

i

⁢

a

⁢

x

/

p

⁢

∑

t

=

0

p

-

1

(

t

p

)

⁢

e

-

2

⁢

π

⁢

i

⁢

a

⁢

t

/

p

The expression

∑

t

=

0

p

-

1

(

t

p

)

⁢

e

-

2

⁢

π

⁢

i

⁢

a

⁢

t

/

p

is just a

Gauss sum

, and has magnitude

p

. Hence

|

∑

t

=

m

m

+

n

(

t

p

)

|

≤

|

p

p

⁢

∑

a

=

1

p

-

1

∑

x

=

m

m

+

n

e

2

⁢

π

⁢

a

⁢

i

⁢

x

/

p

|

=

|

p

p

⁢

∑

a

=

1

p

-

1

e

2

⁢

π

⁢

i

⁢

a

⁢

m

/

p

⁢

∑

x

=

0

n

e

2

⁢

π

⁢

i

⁢

a

⁢

x

/

p

|

≤

|

p

p

⁢

∑

a

=

1

p

-

1

e

2

⁢

π

⁢

i

⁢

a

⁢

n

/

p

-

1

e

2

⁢

π

⁢

i

⁢

a

/

p

-

1

|

=

|

p

p

⁢

∑

a

=

1

p

-

1

e

π

⁢

i

⁢

a

⁢

n

/

p

⁢

sin

⁡

(

π

⁢

a

⁢

n

/

p

)

e

π

⁢

i

⁢

a

/

p

⁢

sin

⁡

(

π

⁢

a

/

p

)

|

≤

p

p

⁢

∑

a

=

1

p

-

1

|

1

sin

⁡

(

π

⁢

⟨

a

/

p

⟩

)

|

≤

p

p

⁢

∑

a

=

1

p

-

1

1

2

⁢

⟨

a

/

p

⟩

Here

⟨

x

⟩

denotes the

absolute value

of the

difference

between

x

and the closest

integer

to

x

, i.e.

⟨

x

⟩

=

inf

z

∈

𝐙

⁡

{

|

x

-

z

|

}

.

Since

p

is odd, we have

1

2

⁢

∑

a

=

1

p

-

1

1

⟨

a

/

p

⟩

=

∑

0

<

a

<

p

2

p

a

=

p

⁢

∑

a

=

1

p

-

1

2

1

a

Now

ln

⁡

2

⁢

x

+

1

2

⁢

x

-

1

>

1

x

for

x

>

1

; to prove this, it
suffices to show that the

function

f

:

[

1

,

∞

)

→

𝐑

given by

f

⁢

(

x

)

=

x

⁢

ln

⁡

2

⁢

x

+

1

2

⁢

x

-

1

is decreasing and approaches 1 as

x

→

∞

. To prove the
latter statement, substitute

v

=

1

/

x

and take the limit as

v

→

0

using L’Hôpital’s rule. To prove the former statement,
it will suffice to show that

f

′

is less than zero on the

interval

[

1

,

∞

)

. But

f

′

⁢

(

x

)

→

0

as

x

→

∞

and

f

′

is increasing on

[

1

,

∞

)

, since

f

′′

⁢

(

x

)

=

-

4

4

⁢

x

2

-

1

⁢

(

1

-

4

⁢

x

2

+

1

4

⁢

x

2

-

1

)

>

0

for

x

>

1

, so

f

′

is less than zero for

x

>

1

.

With this in hand, we have

|

∑

t

=

m

m

+

n

(

t

p

)

|

≤

p

p

⋅

p

⁢

∑

a

=

1

p

-

1

2

1

a

<

p

⁢

∑

a

=

1

p

-

1

2

ln

⁡

2

⁢

a

+

1

2

⁢

a

-

1

=

p

⁢

ln

⁡

p

.

∎

References

1

Vinogradov, I. M.,

Elements of

Number Theory

, 5th rev. ed., Dover, 1954.

Title

Pólya-Vinogradov

inequality

Canonical name

PolyaVinogradovInequality

Date of creation

2013-03-22 12:46:23

Last modified on

2013-03-22 12:46:23

Owner

djao (24)

Last modified by

djao (24)

Numerical id

7

Author

djao (24)

Entry type

Theorem

Classification

msc 11L40

Generated on Fri Feb 9 15:37:04 2018 by

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Apéry’s constant

Apéry’s constant

The number

ζ

⁢

(

3

)

=

∑

n

=

1

∞

1

n

3

=

1.202056903159594285399738161511449990764986292

⁢

…

has been called Apéry’s constant since 1979, when Roger Apéry published
a remarkable proof that it is irrational

[

1

]

.

References

1

Roger Apéry.

Irrationalité de

ζ

⁢

(

2

)

et

ζ

⁢

(

3

)

.

Astérisque

, 61:11–13, 1979.

2

Alfred van der Poorten.

A proof that Euler missed. Apéry’s proof of the irrationality
of

ζ

⁢

(

3

)

. An informal report.

Math. Intell.

, 1:195–203, 1979.

Title

Apéry’s constant

Canonical

name

AperysConstant

Date of creation

2013-03-22 13:27:19

Last modified on

2013-03-22 13:27:19

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Definition

Classification

msc 11M06

Classification

msc 11J81

Generated on Fri Feb 9 15:37:06 2018 by

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convergence of Riemann zeta series

convergence of Riemann zeta series

The series

∑

n

=

1

∞

1

n

s

(1)

converges absolutely

for all

s

with

real part

greater

than 1.

Proof.

Let

s

=

σ

+

i

⁢

t

where

σ

and

t

are

real numbers

and

σ

>

1

.  Then

|

1

n

s

|

=

1

|

e

s

⁢

log

⁡

n

|

=

1

e

σ

⁢

log

⁡

n

=

1

n

σ

.

Since the series

∑

n

=

1

∞

1

n

σ

converges, by the

p

-test (

http://planetmath.org/PTest

), for

σ

>

1

, we conclude that the series (1) is absolutely convergent in the half-plane

σ

>

1

.

Title

convergence of Riemann zeta series

Canonical

name

ConvergenceOfRiemannZetaSeries

Date of creation

2015-08-22 13:15:14

Last modified on

2015-08-22 13:15:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Definition

Classification

msc 11M06

Classification

msc 30A99

Related topic

ModulusOfComplexNumber

Related topic

ComplexExponentialFunction

Generated on Fri Feb 9 15:37:08 2018 by

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critical strip

critical strip

The

critical strip

of the

Riemann zeta function

is

{

s

∈

ℂ

:

0

≤

Re

⁡

(

s

)

≤

1

}

. The zeroes of the Riemann zeta function outside of the critical strip are exactly the set of all

negative

even integers

. The location of the zeroes of the Riemann zeta function inside the critical strip is not totally known; about these zeroes is crucial in

analytic number theory

and the of primes. The

Riemann hypothesis

asserts that all zeroes of the Riemann zeta function that are in the critical strip lie on the line

Re

⁡

(

s

)

=

1

2

. This is all explained in more detail in the entry Riemann zeta function (

http://planetmath.org/RiemannZetaFunction

).

It is well known that no zeroes of the Riemann zeta function lie on either of the lines

Re

⁡

(

s

)

=

0

and

Re

⁡

(

s

)

=

1

. (See this entry (

http://planetmath.org/RiemannZetaFunctionHasNoZerosOnReS01

) for a proof.) Therefore, some people use the “critical strip” to refer to the region (

http://planetmath.org/Region

)

{

s

∈

ℂ

:

0

<

Re

⁡

(

s

)

<

1

}

. (Note that this is the interior of the critical strip as defined above.) For example, this usage in the title of the entry formulae for zeta in the critical strip.

Title

critical strip

Canonical name

CriticalStrip

Date of creation

2013-03-22 16:07:21

Last modified on

2013-03-22 16:07:21

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11M06

Related topic

FormulaeForZetaInTheCriticalStrip

Related topic

ValueOfTheRiemannZetaFunctionAtS0

Related topic

AnalyticContinuationOfRiemannZeta

Generated on Fri Feb 9 15:37:10 2018 by

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Dedekind zeta function

Dedekind zeta function

Let

K

be a

number field

with

ring of integers

𝒪

K

. Then the

Dedekind zeta function

of

K

is the

analytic continuation

of the following series:

ζ

K

⁢

(

s

)

=

∑

I

⊂

𝒪

K

(

N

ℚ

K

⁢

(

I

)

)

-

s

where

I

ranges over non-zero ideals of

𝒪

K

, and

N

ℚ

K

(

I

)

=

|

𝒪

K

:

I

|

is the norm of

I

.

This converges for

ℜ

⁡

(

s

)

>

1

, and has a

meromorphic continuation

to the whole plane, with a

simple pole

at

s

=

1

, and no others.

The Dedekind zeta function has an

Euler product

expansion,

ζ

K

⁢

(

s

)

=

∏

𝔭

1

1

-

(

N

ℚ

K

⁢

(

𝔭

)

)

-

s

where

𝔭

ranges over

prime ideals

of

𝒪

K

. The Dedekind zeta function of

ℚ

is just the

Riemann zeta function

.

Title

Dedekind zeta function

Canonical name

DedekindZetaFunction

Date of creation

2013-03-22 13:20:23

Last modified on

2013-03-22 13:20:23

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

9

Author

bwebste (988)

Entry type

Definition

Classification

msc 11M06

Classification

msc 11R42

Related topic

RiemannZetaFunction

Related topic

ClassNumberFormula

Generated on Fri Feb 9 15:37:12 2018 by

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Dirichlet L-series

Dirichlet L-series

The

Dirichlet L-series

associated to a

Dirichlet character

χ

is the series

L

⁢

(

χ

,

s

)

=

∑

n

=

1

∞

χ

⁢

(

n

)

n

s

.

(1)

It

converges absolutely

and uniformly in the domain

ℜ

⁡

(

s

)

≥

1

+

δ

for any

positive

δ

, and admits the

Euler product

identity

L

⁢

(

χ

,

s

)

=

∏

p

1

1

-

χ

⁢

(

p

)

⁢

p

-

s

(2)

where the product is over all primes

p

, by virtue of the multiplicativity of

χ

. In the case where

χ

=

χ

0

is the trivial character mod m, we have

L

⁢

(

χ

0

,

s

)

=

ζ

⁢

(

s

)

⁢

∏

p

|

m

(

1

-

p

-

s

)

,

(3)

where

ζ

⁢

(

s

)

is the

Riemann Zeta function

. If

χ

is non-primitive, and

C

χ

is the

conductor

of

χ

, we have

L

(

χ

,

s

)

=

L

(

χ

′

,

s

)

∏

p

|

m

p

∤

C

χ

(

1

-

χ

(

p

)

p

-

s

)

,

(4)

where

χ

′

is the

primitive character

which induces

χ

. For non-trivial, primitive characters

χ

mod m,

L

⁢

(

χ

,

s

)

admits an

analytic continuation

to all of

ℂ

and satsfies the symmetric functional equation

L

⁢

(

χ

,

s

)

⁢

(

m

π

)

s

/

2

⁢

Γ

⁢

(

s

+

e

χ

2

)

=

g

1

⁢

(

χ

)

i

e

χ

⁢

m

⁢

L

⁢

(

χ

-

1

,

1

-

s

)

⁢

(

m

π

)

1

-

s

2

⁢

Γ

⁢

(

1

-

s

+

e

χ

2

)

.

(5)

Here,

e

χ

∈

{

0

,

1

}

is defined by

χ

⁢

(

-

1

)

=

(

-

1

)

e

χ

⁢

χ

⁢

(

1

)

,

Γ

is the

gamma function

, and

g

1

⁢

(

χ

)

is a

Gauss sum

.
(3),(4), and (5) combined show that

L

⁢

(

χ

,

s

)

admits a

meromorphic continuation

to all of

ℂ

for all Dirichlet characters

χ

, and an

analytic

one for non-trivial

χ

.
Again assuming that

χ

is non-trivial and primitive character mod m, if

k

is a

positive integer

, we have

L

⁢

(

χ

,

1

-

k

)

=

-

B

k

,

χ

k

,

(6)

where

B

k

,

χ

is a

generalized Bernoulli number

. By (5), taking into account the poles of

Γ

, we get for

k

positive,

k

≡

e

χ

mod 2,

L

⁢

(

χ

,

k

)

=

(

-

1

)

1

+

k

-

e

χ

2

⁢

g

1

⁢

(

χ

)

2

⁢

i

e

χ

⁢

(

2

⁢

π

m

)

k

⁢

B

k

,

χ

-

1

k

!

.

(7)

This series was first investigated by Dirichlet (for whom they were named), who used the non-vanishing of

L

⁢

(

χ

,

1

)

for non-trivial

χ

to prove his famous Dirichlet’s theorem on primes in arithmetic progression. This is probably the first instance of using

complex analysis

to prove a purely number theoretic result.

Title

Dirichlet L-series

Canonical name

DirichletLseries

Date of creation

2013-03-22 13:22:28

Last modified on

2013-03-22 13:22:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11M06

Synonym

Dirichlet L-function

Related topic

LSeriesOfAnEllipticCurve

Related topic

DirichletSeries

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