Splitting field

Template:No footnotes In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of $\mathbb {Z} /k\mathbb {Z}$ . Dirichlet characters are used to define Dirichlet L-functions, which are meromorphic functions with a variety of interesting analytic properties. If $\chi$ is a Dirichlet character, one defines its Dirichlet L-series by

L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}

where s is a complex number with real part > 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. Dirichlet L-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis.

Dirichlet characters are named in honour of Peter Gustav Lejeune Dirichlet.

Axiomatic definition

A Dirichlet character is any function $\chi$ from the integers $\mathbb {Z}$ to the complex numbers $\mathbb {C}$ such that $\chi$ has the following properties:^[1]

There exists a positive integer k such that χ(n) = χ(n + k) for all n.
If gcd(n,k) > 1 then χ(n) = 0; if gcd(n,k) = 1 then χ(n) ≠ 0.
χ(mn) = χ(m)χ(n) for all integers m and n.

From this definition, several other properties can be deduced. By property 3), χ(1)=χ(1×1)=χ(1)χ(1). Since gcd(1, k) = 1, property 2) says χ(1) ≠ 0, so

χ(1) = 1.

Properties 3) and 4) show that every Dirichlet character χ is completely multiplicative.

Property 1) says that a character is periodic with period k; we say that $\chi$ is a character to the modulus k. This is equivalent to saying that

If a ≡ b (mod k) then χ(a) = χ(b).

If gcd(a,k) = 1, Euler's theorem says that a^φ(k) ≡ 1 (mod k) (where φ(k) is the totient function). Therefore by 5) and 4), χ(a^φ(k)) = χ(1) = 1, and by 3), χ(a^φ(k)) =χ(a)^φ(k). So

For all a relatively prime to k, χ(a) is a φ(k)-th complex root of unity.

The unique character of period 1 is called the trivial character. Note that any character vanishes at 0 except the trivial one, which is 1 on all integers.

A character is called principal if it assumes the value 1 for arguments coprime to its modulus and otherwise is 0.^[2] A character is called real if it assumes real values only. A character which is not real is called complex.^[3]

The sign of the character $\chi$ depends on its value at −1. Specifically, $\chi$ is said to be odd if $\chi (-1)=-1$ and even if $\chi (-1)=1$ .

Construction via residue classes

Dirichlet characters may be viewed in terms of the character group of the unit group of the ring Z/kZ, as extended residue class characters.^[4]

Residue classes

Given an integer k, one defines the residue class of an integer n as the set of all integers congruent to n modulo k: ${\hat {n}}=\{m|m\equiv n\mod k\}.$ That is, the residue class ${\hat {n}}$ is the coset of n in the quotient ring Z/kZ.

The set of units modulo k forms an abelian group of order $\phi (k)$ , where group multiplication is given by ${\widehat {mn}}={\hat {m}}{\hat {n}}$ and $\phi$ again denotes Euler's phi function. The identity in this group is the residue class ${\hat {1}}$ and the inverse of ${\hat {m}}$ is the residue class ${\hat {n}}$ where ${\hat {m}}{\hat {n}}={\hat {1}}$ , i.e., $mn\equiv 1\mod k$ . For example, for k=6, the set of units is $\{{\hat {1}},{\hat {5}}\}$ because 0, 2, 3, and 4 are not coprime to 6.

The character group of (Z/k)^* consists of the residue class characters. A residue class character θ on (Z/k)^* is primitive if there is no proper divisor d of k such that θ factors as a map (Z/k)^* → (Z/d)^* → C^*.^[5]

Dirichlet characters

The definition of a Dirichlet character modulo k ensures that it restricts to a character of the unit group modulo k:^[6] a group homomorphism $\chi$ from (Z/kZ)^* to the non-zero complex numbers

\chi :(\mathbb {Z} /k\mathbb {Z} )^{*}\to \mathbb {C} ^{*}

,

with values that are necessarily roots of unity since the units modulo k form a finite group. In the opposite direction, given a group homomorphism $\chi$ on the unit group modulo k, we can lift to a completely multiplicative function on integers relatively prime to k and then extend this function to all integers by defining it to be 0 on integers having a non-trivial factor in common with k. The resulting function will then be a Dirichlet character.^[7]

The principal character $\chi _{1}$ modulo k has the properties^[7]

\chi _{1}(n)=1

if gcd(n, k) = 1 and

\chi _{1}(n)=0

if gcd(n, k) > 1.

The associated character of the multiplicative group (Z/kZ)^* is the principal character which always takes the value 1.^[8]

When k is 1, the principal character modulo k is equal to 1 at all integers. For k greater than 1, the principal character modulo k vanishes at integers having a non-trivial common factor with k and is 1 at other integers.

There are φ(n) Dirichlet characters modulo n.^[7]

A few character tables

The tables below help illustrate the nature of a Dirichlet character. They present all of the characters from modulus 1 to modulus 10. The characters χ₁ are the principal characters.

Modulus 1

There is $\phi (1)=1$ character modulo 1:

χ \ n	0
$\chi _{1}(n)$	1

This is the trivial character.

Modulus 2

There is $\phi (2)=1$ character modulo 2:

χ \ n	0	1
$\chi _{1}(n)$	0	1

Note that χ is wholly determined by χ(1) since 1 generates the group of units modulo 2.

Modulus 3

There are $\phi (3)=2$ characters modulo 3:

χ \ n	1	2
$\chi _{1}(n)$	1	1
$\chi _{2}(n)$	1	−1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 3.

Modulus 4

There are $\phi (4)=2$ characters modulo 4:

χ \ n	1	2	3
$\chi _{1}(n)$	1	0	1
$\chi _{2}(n)$	1	0	−1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 4.

The Dirichlet L-series for $\chi _{1}(n)$ is the Dirichlet lambda function (closely related to the Dirichlet eta function)

L(\chi _{1},s)=(1-2^{-s})\zeta (s)\,

where $\zeta (s)$ is the Riemann zeta-function. The L-series for $\chi _{2}(n)$ is the Dirichlet beta-function

L(\chi _{2},s)=\beta (s).\,

Modulus 5

There are $\phi (5)=4$ characters modulo 5. In the tables, i is a square root of $-1$ .

χ \ n	1	2	3	4
$\chi _{1}(n)$	1	1	1	1
$\chi _{2}(n)$	1	i	−i	−1
$\chi _{3}(n)$	1	−1	−1	1
$\chi _{4}(n)$	1	−i	i	−1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 5.

Modulus 6

There are $\phi (6)=2$ characters modulo 6:

χ \ n	1	2	3	4	5
$\chi _{1}(n)$	1	0	0	0	1
$\chi _{2}(n)$	1	0	0	0	−1

Note that χ is wholly determined by χ(5) since 5 generates the group of units modulo 6.

Modulus 7

There are $\phi (7)=6$ characters modulo 7. In the table below, $\omega =\exp(\pi i/3).$

χ \ n	1	2	3	4	5	6
$\chi _{1}(n)$	1	1	1	1	1	1
$\chi _{2}(n)$	1	ω²	ω	−ω	−ω²	−1
$\chi _{3}(n)$	1	−ω	ω²	ω²	−ω	1
$\chi _{4}(n)$	1	1	−1	1	−1	−1
$\chi _{5}(n)$	1	ω²	−ω	−ω	ω²	1
$\chi _{6}(n)$	1	−ω	−ω²	ω²	ω	−1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 7.

Modulus 8

There are $\phi (8)=4$ characters modulo 8.

χ \ n	1	2	3	4	5	6	7
$\chi _{1}(n)$	1	0	1	0	1	0	1
$\chi _{2}(n)$	1	0	1	0	−1	0	−1
$\chi _{3}(n)$	1	0	−1	0	1	0	−1
$\chi _{4}(n)$	1	0	−1	0	−1	0	1

Note that χ is wholly determined by χ(3) and χ(5) since 3 and 5 generate the group of units modulo 8.

Modulus 9

There are $\phi (9)=6$ characters modulo 9. In the table below, $\omega =\exp(\pi i/3).$

χ \ n	1	2	3	4	5	6	7	8
$\chi _{1}(n)$	1	1	0	1	1	0	1	1
$\chi _{2}(n)$	1	ω	0	ω²	−ω²	0	−ω	−1
$\chi _{3}(n)$	1	ω²	0	−ω	−ω	0	ω²	1
$\chi _{4}(n)$	1	−1	0	1	−1	0	1	−1
$\chi _{5}(n)$	1	−ω	0	ω²	ω²	0	−ω	1
$\chi _{6}(n)$	1	−ω²	0	−ω	ω	0	ω²	−1

Note that χ is wholly determined by χ(2) since 2 generates the group of units modulo 9.

Modulus 10

There are $\phi (10)=4$ characters modulo 10.

χ \ n	1	2	3	4	5	6	7	8	9
$\chi _{1}(n)$	1	0	1	0	0	0	1	0	1
$\chi _{2}(n)$	1	0	i	0	0	0	−i	0	−1
$\chi _{3}(n)$	1	0	−1	0	0	0	−1	0	1
$\chi _{4}(n)$	1	0	−i	0	0	0	i	0	−1

Note that χ is wholly determined by χ(3) since 3 generates the group of units modulo 10.

Examples

If p is an odd prime number, then the function

\chi (n)=\left({\frac {n}{p}}\right),\

where

\left({\frac {n}{p}}\right)

is the Legendre symbol, is a primitive Dirichlet character modulo p.^[9]

More generally, if m is a positive odd number, the function

\chi (n)=\left({\frac {n}{m}}\right),\

where

\left({\frac {n}{m}}\right)

is the Jacobi symbol, is a Dirichlet character modulo m.^[9]

These are quadratic characters: in general, the primitive quadratic characters arise precisely from the Kronecker symbol.^[10]

Primitive characters and conductor

Residues mod N give rise to residues mod M, for any factor M of N, by discarding some information. The effect on Dirichlet characters goes in the opposite direction: if χ is a character mod M, it induces a character χ* mod N for any multiple N of M. A character is primitive if it is not induced by any character of smaller modulus.^[3]

If χ is a character mod n and d divides n, then we say that the modulus d is an induced modulus for χ if a coprime to n and 1 mod d implies χ(a)=1:^[11] equivalently, χ(a) = χ(b) whenever a, b are congruent mod d and each coprime to n.^[12] A character is primitive if there is no smaller induced modulus.^[12]

We can formalise differently this by defining characters χ₁ mod N₁ and χ₂ mod N₂ to be co-trained if for some modulus N such that N₁ and N₂ both divide N we have χ₁(n) = χ₂(n) for all n coprime to N: that is, there is some character χ* induced by each of χ₁ and χ₂. This is an equivalence relation on characters. A character with the smallest modulus in an equivalence class is primitive and this smallest modulus is the conductor of the characters in the class.

Imprimitivity of characters can lead to missing Euler factors in their L-functions.

Character orthogonality

The orthogonality relations for characters of a finite group transfer to Dirichlet characters.^[13] If we fix a character χ modulo n then the sum

\sum _{a{\bmod {n}}}\chi (a)=0\

unless χ is principal, in which case the sum is φ(n). Similarly, if we fix a residue class a modulo n and sum over all characters we have

\sum _{\chi }\chi (a)=0\

unless a=1 in which case the sum is φ(n). We deduce that any periodic function with period n supported on the residue classes prime to n is a linear combination of Dirichlet characters.^[14]

History

Dirichlet characters and their L-series were introduced by Peter Gustav Lejeune Dirichlet, in 1831, in order to prove Dirichlet's theorem on arithmetic progressions. He only studied them for real s and especially as s tends to 1. The extension of these functions to complex s in the whole complex plane was obtained by Bernhard Riemann in 1859.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

See chapter 6 of Template:Apostol IANT
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 see chapter 13.
Template:Cite arxiv
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

my web-site http://himerka.com/

de:Charakter (Mathematik)#Dirichlet-Charaktere

↑ Montgomery & Vaughan (2007) pp.117–8
↑ Montgomery & Vaughan (2007) p.115
↑ ^3.0 ^3.1 Montgomery & Vaughan (2007) p.123
↑ Fröhlich & Taylor (1991) p.218
↑ Frohlich & Taylor (1991) p.215
↑ Apostol (1976) p.139
↑ ^7.0 ^7.1 ^7.2 Apostol (1976) p.138
↑ Apostol (1976) p.134
↑ ^9.0 ^9.1 Montgomery & Vaughan (2007) p.295
↑ Montgomery & Vaughan (2007) p.296
↑ Apostol (1976) p.166
↑ ^12.0 ^12.1 Apostol (1976) p.168
↑ Apostol (1976) p.140
↑ Davenport (1967) pp.31–32

[MV1178-1] Montgomery & Vaughan (2007) pp.117–8

[MV115-2] Montgomery & Vaughan (2007) p.115

[MV123-3] 3.0 ^3.1 Montgomery & Vaughan (2007) p.123

[4] Fröhlich & Taylor (1991) p.218

[FT215-5] Frohlich & Taylor (1991) p.215

[A139-6] Apostol (1976) p.139

[A138-7] 7.0 ^7.1 ^7.2 Apostol (1976) p.138

[A134-8] Apostol (1976) p.134

[MV295-9] 9.0 ^9.1 Montgomery & Vaughan (2007) p.295

[MV296-10] Montgomery & Vaughan (2007) p.296

[A166-11] Apostol (1976) p.166

[A168-12] 12.0 ^12.1 Apostol (1976) p.168

[A140-13] Apostol (1976) p.140

[14] Davenport (1967) pp.31–32

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

Splitting field

Contents

Axiomatic definition