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Seasons in the Sun
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For the Spell album, see Seasons in the Sun (album).
For the 2012 book about British history, see Dominic Sandbrook.
"Le Moribond (The Dying Man)"
Song by Jacques Brel
from the album Marieke
Released
1961
Recorded
February 22, 1961
Genre
Chanson
Length
3:06
Label
Philips
Songwriter(s)
Jacques Brel
Marieke track listing
"Marieke"
(1) "Le Moribond (The Dying Man)"
(2) "Vivre Debout"
(3)
"Seasons in the Sun" is an English-language adaptation of the song "Le Moribond" by Belgian singer-songwriter Jacques Brel[1] with lyrics rewritten by American singer-poet Rod McKuen.[2] It became a worldwide hit in 1974 for Canadian singer Terry Jacks and became a Christmas number one in the UK in 1999 for Westlife. Jacks's version is one of the fewer than forty all-time singles to have sold 10 million copies worldwide.
Contents
[hide]
1 Terry Jacks recording
1.1 Chart performance
1.1.1 Weekly charts
1.1.2 Year-end charts
2 Westlife recording
3 Other versions
4 Parodies and alternate lyrics
5 References
6 External links
Terry Jacks recording[edit]
"Seasons in the Sun"
Single by Terry Jacks
from the album Seasons in the Sun
B-side
"Put the Bone In"
Released
December 1973
Format
7-inch single
Recorded
1973
Genre
Pop
Length
3:24
Label
Bell
Songwriter(s)
Jacques Brel, Rod McKuen
Producer(s)
Terry Jacks
Terry Jacks singles chronology
"Concrete Sea"
(1972) "Seasons in the Sun"
(1973) "If You Go Away"
(1974)
Music video
"Seasons in the Sun" on YouTube
Terry Jacks recorded "Seasons in the Sun" in Vancouver in 1973 with his then-wife Susan Jacks. They made the decision to record the song when The Beach Boys, who had recorded a version with Terry Jacks producing, decided to abandon their recording.
Jacks released his version as a single in 1973 on his own label, Goldfish Records. "Put the Bone In", an original composition about burying a deceased pet dog, was included as the B-side. The single soon topped the record charts in the U.S. (where it was released on Bell Records), in Canada, and the UK,[3] selling over 14 million copies worldwide.
Jacks's version was released in the United States in December 1973, and made the Billboard Hot 100 a month later. On March 2, 1974, the song began a three-week run at number one atop the Hot 100, and remained in the top 40 until almost Memorial Day weekend. Jacks's version also spent one week on the Easy Listening charts.[4] Billboard ranked it as the number two song for 1974.[5] Although he released several other singles that were moderately successful in Canada, "Seasons in the Sun" would become Jacks's only major solo hit in the United States.[6] In Canada, the single (Gold Fish GF 100) reached number one on the RPM Magazine charts January 26, 1974, and remained there four weeks.
Though the song enjoyed contemporary success, modern criticism has tended to be more critical of Jacks's overly sentimentalized rewriting of the original song. Jacks's version is commonly held up as an example of bad music, having been called one of the worst pop songs ever recorded, and ranking number five in a CNN poll in 2006.[7]
Jacks also released a German language version in Germany with lyrics by Gerd Müller-Schwanke, "In den Gärten der Zeit".[8]
Chart performance[edit]
Weekly charts[edit]
Chart (1973–74)
Peak
position
Australia KMR[9] 1
Austria (Ö3 Austria Top 40)[10] 1
Belgium[11] 1
Canada RPM Top Singles[11] 1
Canada RPM Adult Contemporary 1
Denmark[11] 1
France[11] 1
Germany[11] 1
Ireland[11] 1
Netherlands (Single Top 100)[12] 7
Norway (VG-lista)[13] 1
South Africa[11] 1
Switzerland (Schweizer Hitparade)[14] 1
New Zealand[15] 1
UK Singles (Official Charts Company)[16] 1
US Billboard Hot 100[17] 1
US Adult Contemporary (Billboard)[18] 1
US Cash Box Top 100[19] 1
Year-end charts[edit]
Chart (1974)
Rank
Australia[20] 5
Canada RPM Top Singles[21] 1
Denmark[11] 5
Netherlands[11] 6
South Africa[11] 6
Switzerland[22] 1
UK[23] 5
U.S. Billboard Hot 100[24] 2
U.S. Cash Box[25] 9
Westlife recording[edit]
"Seasons in the Sun"
Single by Westlife
from the album Westlife
A-side
"I Have a Dream"
Released
December 19, 1999
Format
CD single
Studio
PWL Studios, London
Genre
Pop
Length
4:09
Label
BMG
Songwriter(s)
Jacques Brel, Rod McKuen
Westlife singles chronology
"Flying Without Wings"
(1999) "Seasons in the Sun"
(1999) "Fool Again"
(2000)
Music video
"Seasons in the Sun" on YouTube
Irish boy band Westlife covered "Seasons in the Sun" for their self-titled debut album Westlife (1999). It was released on December 19, 1999, as a double A-side with a cover of ABBA's "I Have a Dream" and became the 1999 Christmas number one. It has sold over 650,000 copies in UK so far.[26]
Westlife's "Seasons In The Sun" music video was shot in late-1999 and directed by Cameron Casey & Simon Brand.
Other versions[edit]
There have been numerous cover versions of the song. Before Jacks popularized the song, earlier recordings had been released by The Kingston Trio, with the first cover version of McKuen's translation in 1963, and The Fortunes in 1968.
Generally, English-language versions since the Terry Jacks version was released have used his lyrics for the song.
Swedish dansband Vikingarna covered the song in Swedish in 1974, as "Sommar varje dag" on the album Här kommer Vikingarna.[27] The song has also been covered by Andy Williams, The Beach Boys, Nana Mouskouri, Bud Dashiell, Bad Religion, Too Much Joy, the Squirrels, Spell, Black Box Recorder, Nirvana, Indochine, Karel Gott (Léta prázdnin), Klaus Hoffmann (in German), Pearls Before Swine, Alcazar, Me First and the Gimme Gimmes, Second Run, The Others, Television Personalities, and Gob featuring blink-182.
In 1993, Nirvana recorded an informal version of the song which was never intended for release, but was eventually included on the DVD portion of their 2004 box-set With the Lights Out. Nirvana's lead singer, Kurt Cobain, would often reference the song in interviews, mentioning the fact that it made him cry as a child.
Another cover version, by Bobby Wright (son of Johnnie Wright and Kitty Wells), reached the Top 40 of the Billboard magazine Hot Country Singles chart in 1974.
The New York-based band, Too Much Joy, covered the song on their album Son of Sam I Am, Alias Records (1989), Warner Bros./Giant Records (1990).[28][better source needed]
The Manchester-based Coachmen's version was recorded at EMI's Abbey Road Studios in July 1966 and released on Columbia DB8057 in November 1966.
Spanish versions are entitled 'Epocas de Sol, Estaciones en el Sol, or Etapas de mi Vida.
There is a German version called Adieu Emile by Klaus Hoffmann published in 1975. The translated lyrics are very close to the French ones but the melody is slightly altered and the whole song has a 3/4 beat.[29]
Italian singer/songwriter Roberto Vecchioni recorded an Italian version called "Stagioni nel sole" for his 2005 acoustic, jazz-influenced live album Il Contastorie. The title of his version is a literal translation of "Seasons in the Sun", but both the lyrics (rewritten by Vecchioni himself) and the musical structure of Vecchioni's version are much closer to Jacques Brel's original (including a vague allusion to the protagonist's wife being unfaithful) than to the Jacks/McKuen version. The accompaniment of this rendition, played on acoustic guitar, piano and double bass (with no drums) is march-like, similar to Brel's, and pointedly un-romantic.
A second French-language version, Adieu la Vie, Adieu Soleil, was recorded in 1974 by Quebec musician Jacques Amar; this version is the Québécois adaptation of Terry Jacks' version.[30]
Croatian singer-songwriter Arsen Dedić recorded a Croatian version of the song in 1983 under the title Pjesma umirućeg. Dedić's lyrics draw more heavily on McKuen's English translation.
The band Beirut has covered "Le Moribond" live on a number of occasions. A version performed for KEXP was recorded along with "My Family's Role in the World Revolution" for the third track on the Elephant Gun EP. Beirut performs a cover that is quite faithful to the original Jacques Brel version, maintaining the French lyrics and the marching up-beat tempo.
The Finnish schlager singer Arto Sotavalta have recorded a Finnish version of the song called Päivät kuin unta (Eng. Days like dreams). The Finnish translation is very similar to the English text.
The Vietnamese version, called "Những mùa nắng đẹp" was translated by musician Pham Duy. The translated lyrics are quite similar to the English ones. A number of Vietnamese singers have covered this version.
Swedish Eurodance group Alcazar covered this song for their debut album Casino.
British musician Damon Albarn recorded an acoustic version[31] of this song for a BBC Radio 2 show with Dermot O'Leary.
Malaysia's pop artist Sharifah Aini also released her own version of the song called Cahaya Keinsafan.[32]
Parodies and alternate lyrics[edit]
The 1993 song "Twice My Age" by Shabba Ranks uses the melody of "Seasons in the Sun" with a different set of lyrics. Sean Kingston covered "Twice My Age" on his 2011 mixtape King of Kingz.
In 2002 a satirical version of the song lambasting the Canadian government's ageing Sea King helicopters was leaked to Esprit de Corps magazine. The lyrics were published in the Ottawa Citizen on November 20, 2002.
Another spoof was recorded in 2006 by Dundee United fan under the alias 'Terry Jack'. This version mocks the possible demise of Dundee United's archrivals, Dundee FC who were suffering severe financial difficulties at the time.[33] This version contains swearing and adult themes.
References[edit]
Jump up ^ "30 years since the death of Jacques Brel: his life, his art, his legacy - World Socialist Web Site". Wsws.org. Retrieved 2014-03-27.
Jump up ^ "My first experience with Brel (...in 1964...) consisted of doing an unauthorized adaptation of 'Le Moribund'(...written by Brel for the singer Jean Sablon...), which I called 'Seasons in the Sun'...I subsequently learned that Brel had received my recording...," liner notes from Rod McKuen Sings Rod McKuen, Jacques Brel Songbook, April 2000, posted at Rod McKuen - Flight Plan. 'Rod's gift to his many fans', maintained by Stanyan Music Group, of Hollywood CA. Accessed: 12 September 2009.
Jump up ^ Roberts, David (2006). British Hit Singles & Albums (19th ed.). London: Guinness World Records Limited. p. 299. ISBN 1-904994-10-5.
Jump up ^ Whitburn, Joel (2002). Top Adult Contemporary: 1961-2001. Record Research. p. 123.
Jump up ^ "Number One Song of the Year: 1946-2015". bobborst.com. Retrieved August 22, 2016.
Jump up ^ Whitburn, Joel, "Top Pop Hits: 1955-2006, 2007.
Jump up ^ Todd Leopold. "The worst song of all time, part II - Apr 26, 2006". CNN.com. Retrieved 2014-02-01.
Jump up ^ "In den Gärten der Zeit" at 45cat.com
Jump up ^ "Australian Chart Book". Austchartbook.com.au. Retrieved 2016-10-08.
Jump up ^ "Austriancharts.at – TERRY JACKS – SEASONS IN THE SUN" (in German). Ö3 Austria Top 40.
^ Jump up to: a b c d e f g h i j "Song artist 781 - Terry Jacks". Tsort.info. Retrieved 2016-10-08.
Jump up ^ "Dutchcharts.nl – TERRY JACKS – SEASONS IN THE SUN" (in Dutch). Single Top 100.
Jump up ^ "Norwegiancharts.com – TERRY JACKS – SEASONS IN THE SUN". VG-lista.
Jump up ^ "Swisscharts.com – TERRY JACKS – SEASONS IN THE SUN". Swiss Singles Chart.
Jump up ^ "flavour of new zealand - search listener". Flavourofnz.co.nz. Retrieved 2016-10-08.
Jump up ^ "Terry Jacks: Artist Chart History". Official Charts Company.
Jump up ^ "Terry Jacks Chart History (Hot 100)". Billboard.
Jump up ^ "Terry Jacks Chart History (Adult Contemporary)". Billboard.
Jump up ^ "Cash Box Top 100 3/09/74". Tropicalglen.com. 1974-03-09. Retrieved 2016-10-08.
Jump up ^ Steffen Hung. "Forum - Top 100 End of Year AMR Charts - 1980s (ARIA Charts: Special Occasion Charts)". Australian-charts.com. Retrieved 2016-10-08.
Jump up ^ "Image : RPM Weekly - Library and Archives Canada". Bac-lac.gc.ca. Retrieved 2016-10-08.
Jump up ^ Hung, Steffen. "Swiss Year-End Charts 1974 - swisscharts.com". swisscharts.com. Retrieved 8 April 2018.
Jump up ^ "Top 100 1974 - UK Music Charts". Uk-charts.top-source.info. Retrieved 2016-10-08.
Jump up ^ "Top 100 Hits of 1974/Top 100 Songs of 1974". Musicoutfitters.com. Retrieved 2016-10-08.
Jump up ^ "Cash Box YE Pop Singles - 1974". Tropicalglen.com. 1974-12-28. Retrieved 2016-10-08.
Jump up ^ "Westlife | Official Top 20 | MTV UK". Mtv.co.uk. 2009-10-28. Retrieved 2014-03-27.
Jump up ^ "Här kommer Vikingarna" (in Swedish). Svensk mediedatabas. 1974. Retrieved 7 May 2011.
Jump up ^ Too Much Joy#Son of Sam I Am
Jump up ^ "Klaus Hoffmann". Klaus Hoffmann. 2007-07-01. Retrieved 2014-03-27.
Jump up ^ "JACQUES AMAR.....ADIEU LA VIE ADIEU SOLEIL (SEASONS IN THE SUN)...1974" on YouTube
Jump up ^ "Damon Albarn covers 'Seasons In The Sun'". NME.COM. Retrieved 2016-05-14.
Jump up ^ "Lagu Saduran dalam Industri Hiburan". www.100penulis.com. Retrieved 8 April 2018.
Jump up ^ [1] Archived 2012-12-01 at the Wayback Machine.
External links[edit]
Listen to this article (info/dl)
This audio file was created from a revision of the article "Seasons in the Sun" dated 2005-12-10, and does not reflect subsequent edits to the article. (Audio help)
More spoken articles
Seasons in the Sun at Super Seventies.
"List of 'Le Moribond' covers", Brelitude.net
"Goodbye, Papa, It's Hard to Die: The enduring appeal of an abominable pop song" (Slate.com, March 16, 2005)
Rod McKuen on the song's origins
Lyrics of Andy Williams cover at MetroLyrics
Preceded by
"Time in a Bottle" by Jim Croce Canadian RPM 100 number-one single
(Terry Jacks's version)
January 26, 1974 (four weeks) Succeeded by
"The Way We Were" by Barbra Streisand
Preceded by
"The Way We Were" by Barbra Streisand US Billboard Hot 100 number one single
(Terry Jacks's version)
March 2, 1974 (three weeks) Succeeded by
"Dark Lady" by Cher
Preceded by
"Last Time I Saw Him" by Diana Ross US Billboard Easy Listening Singles number-one single
(Terry Jacks's version)
March 9, 1974 (one week) Succeeded by
"Sunshine on My Shoulders" by John Denver
Preceded by
"Billy Don't Be a Hero" by Paper Lace UK Singles Chart number one single
(Terry Jacks's version)
April 6, 1974 (four weeks) Succeeded by
"Waterloo" by ABBA
Preceded by
"Goodbye" by Spice Girls UK Singles Chart Christmas number-one single
(Westlife's version)
1999 Succeeded by
"Can We Fix It?" by Bob the Builder
Preceded by
"I Try" by Macy Gray Irish IRMA number one single
(Westlife's version)
18 December 1999 (five weeks) Succeeded by
"Born to Make You Happy" by Britney Spears
Preceded by
"The Millennium Prayer" by Cliff Richard UK Singles Chart number-one single
(Westlife's version)
19 December 1999 (four weeks) Succeeded by
"The Masses Against the Classes" by Manic Street Preachers
[show]
v
t
e
Jacques Brel (discography)
Studio albums
Grand Jacques
Quand on n'a que l'amour
Au Printemps
La Valse à Mille Temps
Marieke
Les Bourgeois
Les Bonbons
Ces Gens-Là
Jacques Brel 67
J'arrive
L'Homme de la Mancha
Ne Me Quitte Pas
Les Marquises
Live albums
Enregistrement Public à l'Olympia 1961
Enregistrement Public à l'Olympia 1964
Compilation albums
Infiniment
Rarities
Chansons ou Versions Inédites de Jeunesse
Box sets
Boîte à Bonbons
Songs
"Ne me quitte pas" ("If You Go Away")
"Marieke"
"Seasons in the Sun"
"Le Plat Pays"
"Ces Gens-Là"
"Amsterdam"
"The Impossible Dream"
"Un enfant"
"Jacky"
Films
Les risques du métier
La bande à Bonnot
Mon oncle Benjamin
Mont-Dragon
Law Breakers
Franz (film)
L'aventure c'est l'aventure
The Bar at the Crossing
Le Far West
L'emmerdeur
Related articles
Calvary Cemetery (Atuona)
L'Envol (statue)
Hiva Oa
Jacques Brel is Alive and Well and Living in Paris (musical)
Jacques Brel Is Alive and Well and Living in Paris (film)
Man of La Mancha
Marieke (statue)
List of cover versions
François Rauber
[show]
v
t
e
Westlife singles
Discography
Westlife
"Swear It Again"
"If I Let You Go"
"Flying Without Wings"
"I Have a Dream"/"Seasons in the Sun"
"Fool Again"
Coast to Coast
"Against All Odds"
"My Love"
"What Makes a Man"
"I Lay My Love on You"
"Uptown Girl"
"When You're Looking Like That"
World of Our Own
"Queen of My Heart"
"Evergreen"
"World of Our Own"
"Bop Bop Baby"
Unbreakable
"Unbreakable"
"Tonight/Miss You Nights"
Turnaround
"Hey Whatever"
"Mandy"
"Obvious"
Allow Us to Be Frank
"Ain't That a Kick in the Head"
"Fly Me to the Moon"
"Smile"
Face to Face
"You Raise Me Up"
"When You Tell Me That You Love Me"
"Amazing"
The Love Album
"The Rose"
"Total Eclipse of the Heart"
Back Home
"Home"
"Us Against the World"
"Something Right"
Where We Are
"What About Now"
Gravity
"Safe"
Greatest Hits
"Lighthouse"
"Beautiful World"
Book
Category
[show]
v
t
e
UK Christmas number-one singles in the 1990s
"Saviour's Day" (Cliff Richard, 1990)
"Bohemian Rhapsody" / "These Are the Days of Our Lives" (Queen, 1991)
"I Will Always Love You" (Whitney Houston, 1992)
"Mr Blobby" (Mr Blobby, 1993)
"Stay Another Day" (East 17, 1994)
"Earth Song" (Michael Jackson, 1995)
"2 Become 1" (Spice Girls, 1996)
"Too Much" (Spice Girls, 1997)
"Goodbye" (Spice Girls, 1998)
"I Have a Dream" / "Seasons in the Sun" (Westlife, 1999)
Complete list
1950s
1960s
1970s
1980s
1990s
2000s
2010s
Retrieved from "https://en.wikipedia.org/w/index.php?title=Seasons_in_the_Sun&oldid=837764191"
Categories:
Christmas number-one singles in the United Kingdom
1961 songs
1973 singles
1974 singles
1999 singles
2000 singles
Songs written by Jacques Brel
Songs written by Rod McKuen
Jacques Brel songs
The Kingston Trio songs
Vikingarna (band) songs
Westlife songs
Nirvana (band) songs
Songs about death
Canadian pop songs
Bobby Wright songs
Andy Williams songs
Billboard Hot 100 number-one singles
Billboard Adult Contemporary number-one singles
Number-one singles in Australia
Number-one singles in Germany
Irish Singles Chart number-one singles
Number-one singles in New Zealand
Number-one singles in Norway
Number-one singles in Switzerland
UK Singles Chart number-one singles
Bell Records singles
Philips Records singles
Pop ballads
Juno Award for Single of the Year singles
Hidden categories:
CS1 Swedish-language sources (sv)
Webarchive template wayback links
Articles with hAudio microformats
Singlechart usages for Austria
Singlechart usages for Dutch100
Singlechart usages for Norway
Singlechart usages for Switzerland
Singlechart usages for Billboardhot100
Singlechart called without song
Singlechart usages for Billboardadultcontemporary
All articles lacking reliable references
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Euler's totient function
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"Φ(n)" redirects here. For other uses, see phi.
The first thousand values of φ(n). The points on the top line represent φ(p) when p is a prime number, which is p − 1.[1]
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as φ(n) or ϕ(n), and may also be called Euler's phi function. It can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1.[2][3] The integers k of this form are sometimes referred to as totatives of n.
For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, because gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1.
Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n).[4][5] This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring ℤ/nℤ).[6] It also plays a key role in the definition of the RSA encryption system.
Contents
[hide]
1 History, terminology, and notation
2 Computing Euler's totient function
2.1 Euler's product formula
2.1.1 The function is multiplicative
2.1.2 Value for a prime power argument
2.1.3 Proof of Euler's product formula
2.1.4 Example
2.2 Fourier transform
2.3 Divisor sum
2.4 Riemann zeta function limit
3 Some values of the function
4 Euler's theorem
5 Other formulae
5.1 Menon's identity
5.2 Formulae involving the golden ratio
6 Generating functions
7 Growth rate
8 Ratio of consecutive values
9 Totient numbers
9.1 Ford's theorem
9.2 Perfect totient numbers
10 Applications
10.1 Cyclotomy
10.2 The RSA cryptosystem
11 Unsolved problems
11.1 Lehmer's conjecture
11.2 Carmichael's conjecture
12 See also
13 Notes
14 References
15 External links
History, terminology, and notation[edit]
Leonhard Euler introduced the function in 1763.[7][8][9] However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it: he wrote πD for "the multitude of numbers less than D, and which have no common divisor with it".[10] This definition varies from the current definition for the totient function at D = 1 but is otherwise the same. The now-standard notation[8][11] φ(A) comes from Gauss's 1801 treatise Disquisitiones Arithmeticae.[12] although Gauss didn't use parentheses around the argument and wrote φA. Thus, it is often called Euler's phi function or simply the phi function.
In 1879, J. J. Sylvester coined the term totient for this function,[13][14] so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's.
The cototient of n is defined as n − φ(n). It counts the number of positive integers less than or equal to n that have at least one prime factor in common with n.
Computing Euler's totient function[edit]
There are several formulas for computing φ(n).
Euler's product formula[edit]
It states
φ ( n ) = n ∏ p ∣ n ( 1 − 1 p ) , {\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right),}
where the product is over the distinct prime numbers dividing n. (The notation is described in the article Arithmetical function.)
The proof of Euler's product formula depends on two important facts.
The function is multiplicative[edit]
This means that if gcd(m, n) = 1, then φ(mn) = φ(m) φ(n). (Outline of proof: let A, B, C be the sets of nonnegative integers, which are, respectively, coprime to and less than m, n, and mn; then there is a bijection between A × B and C, by the Chinese remainder theorem.)
Value for a prime power argument[edit]
If p is prime and k ≥ 1, then
φ ( p k ) = p k − p k − 1 = p k ( 1 − 1 p ) . {\displaystyle \varphi (p^{k})=p^{k}-p^{k-1}=p^{k}\left(1-{\frac {1}{p}}\right).}
Proof: since p is a prime number the only possible values of gcd(pk, m) are 1, p, p2, ..., pk, and the only way for gcd(pk, m) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to pk are p, 2p, 3p, ..., pk − 1p = pk, and there are pk − 1 of them. Therefore, the other pk − pk − 1 numbers are all relatively prime to pk.
Proof of Euler's product formula[edit]
The fundamental theorem of arithmetic states that if n > 1 there is a unique expression for n,
n = p 1 k 1 ⋯ p r k r , {\displaystyle n={p_{1}}^{k_{1}}\cdots {p_{r}}^{k_{r}},}
where p1 < p2 < ... < pr are prime numbers and each ki ≥ 1. (The case n = 1 corresponds to the empty product.)
Repeatedly using the multiplicative property of φ and the formula for φ(pk) gives
φ ( n ) = φ ( p 1 k 1 ) φ ( p 2 k 2 ) ⋯ φ ( p r k r ) = p 1 k 1 ( 1 − 1 p 1 ) p 2 k 2 ( 1 − 1 p 2 ) ⋯ p r k r ( 1 − 1 p r ) = p 1 k 1 p 2 k 2 ⋯ p r k r ( 1 − 1 p 1 ) ( 1 − 1 p 2 ) ⋯ ( 1 − 1 p r ) = n ( 1 − 1 p 1 ) ( 1 − 1 p 2 ) ⋯ ( 1 − 1 p r ) . {\displaystyle {\begin{aligned}\varphi (n)&=\varphi \left(p_{1}^{k_{1}}\right)\varphi \left(p_{2}^{k_{2}}\right)\cdots \varphi \left(p_{r}^{k_{r}}\right)\\&=p_{1}^{k_{1}}\left(1-{\frac {1}{p_{1}}}\right)p_{2}^{k_{2}}\left(1-{\frac {1}{p_{2}}}\right)\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{r}}}\right)\\&=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right)\\&=n\left(1-{\frac {1}{p_{1}}}\right)\left(1-{\frac {1}{p_{2}}}\right)\cdots \left(1-{\frac {1}{p_{r}}}\right).\end{aligned}}}
This is Euler's product formula.
Example[edit]
φ ( 36 ) = φ ( 2 2 3 2 ) = 36 ( 1 − 1 2 ) ( 1 − 1 3 ) = 36 ⋅ 1 2 ⋅ 2 3 = 12. {\displaystyle \varphi (36)=\varphi \left(2^{2}3^{2}\right)=36\left(1-{\tfrac {1}{2}}\right)\left(1-{\tfrac {1}{3}}\right)=36\cdot {\tfrac {1}{2}}\cdot {\tfrac {2}{3}}=12.}
In words, this says that the distinct prime factors of 36 are 2 and 3; half of the thirty-six integers from 1 to 36 are divisible by 2, leaving eighteen; a third of those are divisible by 3, leaving twelve numbers that are coprime to 36. And indeed there are twelve positive integers that are coprime with 36 and lower than 36: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.
Fourier transform[edit]
The totient is the discrete Fourier transform of the gcd, evaluated at 1.[15] Let
F { x } [ m ] = ∑ k = 1 n x k ⋅ e − 2 π i m k n {\displaystyle {\mathcal {F}}\{\mathbf {x} \}[m]=\sum \limits _{k=1}^{n}x_{k}\cdot e^{{-2\pi i}{\frac {mk}{n}}}}
where xk = gcd(k,n) for k ∈ {1, …, n}. Then
φ ( n ) = F { x } [ 1 ] = ∑ k = 1 n gcd ( k , n ) e − 2 π i k n . {\displaystyle \varphi (n)={\mathcal {F}}\{\mathbf {x} \}[1]=\sum \limits _{k=1}^{n}\gcd(k,n)e^{-2\pi i{\frac {k}{n}}}.}
The real part of this formula is
φ ( n ) = ∑ k = 1 n gcd ( k , n ) cos  2 π k n . {\displaystyle \varphi (n)=\sum \limits _{k=1}^{n}\gcd(k,n)\cos {2\pi {\frac {k}{n}}}.}
Note that unlike the other two formulae (the Euler product and the divisor sum) this one does not require knowing the factors of n. However, it does involve the calculation of the greatest common divisor of n and every positive integer less than n, which suffices to provide the factorization anyway.
Divisor sum[edit]
The property established by Gauss,[16] that
∑ d ∣ n φ ( d ) = n , {\displaystyle \sum _{d\mid n}\varphi (d)=n,}
where the sum is over all positive divisors d of n, can be proven in several ways. (see Arithmetical function for notational conventions.)
One way is to note that φ(d) is also equal to the number of possible generators of the cyclic group Cd; specifically, if Cd = ⟨g⟩, then gk is a generator for every k coprime to d. Since every element of Cn generates a cyclic subgroup, and all subgroups of Cd ≤ Cn are generated by some element of Cn, the formula follows.[17] In the article Root of unity Euler's formula is derived by using this argument in the special case of the multiplicative group of the nth roots of unity.
This formula can also be derived in a more concrete manner.[18] Let n = 20 and consider the fractions between 0 and 1 with denominator 20:
1 20 , 2 20 , 3 20 , 4 20 , 5 20 , 6 20 , 7 20 , 8 20 , 9 20 , 10 20 , 11 20 , 12 20 , 13 20 , 14 20 , 15 20 , 16 20 , 17 20 , 18 20 , 19 20 , 20 20 {\displaystyle {\tfrac {1}{20}},\,{\tfrac {2}{20}},\,{\tfrac {3}{20}},\,{\tfrac {4}{20}},\,{\tfrac {5}{20}},\,{\tfrac {6}{20}},\,{\tfrac {7}{20}},\,{\tfrac {8}{20}},\,{\tfrac {9}{20}},\,{\tfrac {10}{20}},\,{\tfrac {11}{20}},\,{\tfrac {12}{20}},\,{\tfrac {13}{20}},\,{\tfrac {14}{20}},\,{\tfrac {15}{20}},\,{\tfrac {16}{20}},\,{\tfrac {17}{20}},\,{\tfrac {18}{20}},\,{\tfrac {19}{20}},\,{\tfrac {20}{20}}}
Put them into lowest terms:
1 20 , 1 10 , 3 20 , 1 5 , 1 4 , 3 10 , 7 20 , 2 5 , 9 20 , 1 2 , 11 20 , 3 5 , 13 20 , 7 10 , 3 4 , 4 5 , 17 20 , 9 10 , 19 20 , 1 {\displaystyle {\tfrac {1}{20}},\,{\tfrac {1}{10}},\,{\tfrac {3}{20}},\,{\tfrac {1}{5}},\,{\tfrac {1}{4}},\,{\tfrac {3}{10}},\,{\tfrac {7}{20}},\,{\tfrac {2}{5}},\,{\tfrac {9}{20}},\,{\tfrac {1}{2}},\,{\tfrac {11}{20}},\,{\tfrac {3}{5}},\,{\tfrac {13}{20}},\,{\tfrac {7}{10}},\,{\tfrac {3}{4}},\,{\tfrac {4}{5}},\,{\tfrac {17}{20}},\,{\tfrac {9}{10}},\,{\tfrac {19}{20}},\,1}
First note that all the divisors of 20 are denominators. And second, note that there are 20 fractions. Which fractions have 20 as denominator? The ones whose numerators are relatively prime to 20 (1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20). By definition this is φ(20) fractions. Similarly, there are φ(10) = 4 fractions with denominator 10 (1/10, 3/10, 7/10, 9/10), φ(5) = 4 fractions with denominator 5 (1/5, 2/5, 3/5, 4/5), and so on.
In detail, we are considering the fractions of the form k/n where k is an integer from 1 to n inclusive. Upon reducing these to lowest terms, each fraction will have as its denominator some divisor of n. We can group the fractions together by denominator, and we must show that for a given divisor d of n, the number of such fractions with denominator d is φ(d).
Note that to reduce k/n to lowest terms, we divide the numerator and denominator by gcd(k, n). The reduced fractions with denominator d are therefore precisely the ones originally of the form k/n in which gcd(k, n) = n/d. The question therefore becomes: how many k are there less than or equal to n which verify gcd(k, n) = n/d? Any such k must clearly be a multiple of n/d, but it must also be coprime to d (if it had any common divisor s with d, then sn/d would be a larger common divisor of n and k). Conversely, any multiple k of n/d which is coprime to d will satisfy gcd(k, n) = n/d. We can generate φ(d) such numbers by taking the numbers less than d coprime to d and multiplying each one by n/d (these products will of course each be smaller than n, as required). This in fact generates all such numbers, as if k is a multiple of n/d coprime to d (and less than n), then k/n⁄d will still be coprime to d, and must also be smaller than d, else k would be larger than n. Thus there are precisely φ(d) values of k less than or equal to n such that gcd(k, n) = n/d, which was to be demonstrated.
Möbius inversion gives
φ ( n ) = ∑ d ∣ n μ ( d ) ⋅ n d = n ∑ d ∣ n μ ( d ) d , {\displaystyle \varphi (n)=\sum _{d\mid n}\mu \left(d\right)\cdot {\frac {n}{d}}=n\sum _{d\mid n}{\frac {\mu (d)}{d}},}
where μ is the Möbius function.
This formula may also be derived from the product formula by multiplying out
∏ p ∣ n ( 1 − 1 p ) {\displaystyle \prod _{p\mid n}\left(1-{\frac {1}{p}}\right)}
to get
∑ d ∣ n μ ( d ) d . {\displaystyle \sum _{d\mid n}{\frac {\mu (d)}{d}}.}
Riemann zeta function limit[edit]
For n > 1 the Euler totient function can be calculated as a limit involving the Riemann zeta function:[citation needed]
φ ( n ) = n lim s → 1 ζ ( s ) ∑ d ∣ n μ ( d ) ( e 1 d ) s − 1 {\displaystyle \varphi (n)=n\lim \limits _{s\rightarrow 1}\zeta (s)\sum \limits _{d\mid n}\mu (d)\left(e^{\frac {1}{d}}\right)^{s-1}}
where ζ(s) is the Riemann zeta function, μ is the Möbius function and d is a divisor.
Some values of the function[edit]
The first 99 values (sequence A000010 in the OEIS) are shown in the table and graph below:[19]
Graph of the first 100 values
φ(n) for 1 ≤ n ≤ 99
+
0
1
2
3
4
5
6
7
8
9
0
N/A 1 1 2 2 4 2 6 4 6
10
4 10 4 12 6 8 8 16 6 18
20
8 12 10 22 8 20 12 18 12 28
30
8 30 16 20 16 24 12 36 18 24
40
16 40 12 42 20 24 22 46 16 42
50
20 32 24 52 18 40 24 36 28 58
60
16 60 30 36 32 48 20 66 32 44
70
24 70 24 72 36 40 36 60 24 78
80
32 54 40 82 24 64 42 56 40 88
90
24 72 44 60 46 72 32 96 42 60
The top line in the graph, y = n − 1, is a true upper bound. It is attained whenever n is prime. There is no lower bound that is a straight line of positive slope; no matter how gentle the slope of a line is, there will eventually be points of the plot below the line. More precisely, the lower limit of the graph is proportional to n/log log n rather than being linear.[20]
Euler's theorem[edit]
Main article: Euler's theorem
This states that if a and n are relatively prime then
a φ ( n ) ≡ 1 mod n . {\displaystyle a^{\varphi (n)}\equiv 1\mod n.}
The special case where n is prime is known as Fermat's little theorem.
This follows from Lagrange's theorem and the fact that φ(n) is the order of the multiplicative group of integers modulo n.
The RSA cryptosystem is based on this theorem: it implies that the inverse of the function a ↦ ae mod n, where e is the (public) encryption exponent, is the function b ↦ bd mod n, where d, the (private) decryption exponent, is the multiplicative inverse of e modulo φ(n). The difficulty of computing φ(n) without knowing the factorization of n is thus the difficulty of computing d: this is known as the RSA problem which can be solved by factoring n. The owner of the private key knows the factorization, since an RSA private key is constructed by choosing n as the product of two (randomly chosen) large primes p and q. Only n is publicly disclosed, and given the difficulty to factor large numbers we have the guarantee that no-one else knows the factorization.
Other formulae[edit]
a ∣ b ⟹ φ ( a ) ∣ φ ( b ) {\displaystyle a\mid b\implies \varphi (a)\mid \varphi (b)}
n ∣ φ ( a n − 1 ) for a , n > 1 {\displaystyle n\mid \varphi (a^{n}-1)\quad {\text{for }}a,n>1}
φ ( m n ) = φ ( m ) φ ( n ) ⋅ d φ ( d ) where d = gcd  ( m , n ) {\displaystyle \varphi (mn)=\varphi (m)\varphi (n)\cdot {\frac {d}{\varphi (d)}}\quad {\text{where }}d=\operatorname {gcd} (m,n)}
Note the special cases
φ ( 2 m ) = { 2 φ ( m ) if m is even φ ( m ) if m is odd {\displaystyle \varphi (2m)={\begin{cases}2\varphi (m)&{\text{ if }}m{\text{ is even}}\\\varphi (m)&{\text{ if }}m{\text{ is odd}}\end{cases}}}
φ ( n m ) = n m − 1 φ ( n ) {\displaystyle \varphi \left(n^{m}\right)=n^{m-1}\varphi (n)}
φ ( lcm  ( m , n ) ) ⋅ φ ( gcd  ( m , n ) ) = φ ( m ) ⋅ φ ( n ) {\displaystyle \varphi (\operatorname {lcm} (m,n))\cdot \varphi (\operatorname {gcd} (m,n))=\varphi (m)\cdot \varphi (n)}
Compare this to the formula
lcm  ( m , n ) ⋅ gcd  ( m , n ) = m ⋅ n {\displaystyle \operatorname {lcm} (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n}
(See least common multiple.)
φ(n) is even for n ≥ 3. Moreover, if n has r distinct odd prime factors, 2r | φ(n)
For any a > 1 and n > 6 such that 4 ∤ n there exists an l ≥ 2n such that l | φ(an − 1).
φ ( n ) n = φ ( rad  ( n ) ) rad  ( n ) {\displaystyle {\frac {\varphi (n)}{n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}}
where rad(n) is the radical of n.
∑ d ∣ n μ 2 ( d ) φ ( d ) = n φ ( n ) {\displaystyle \sum _{d\mid n}{\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}} [21]
∑ 1 ≤ k ≤ n ( k , n ) = 1 k = 1 2 n φ ( n ) for n > 1 {\displaystyle \sum _{1\leq k\leq n \atop (k,n)=1}\!\!k={\tfrac {1}{2}}n\varphi (n)\quad {\text{for }}n>1}
∑ k = 1 n φ ( k ) = 1 2 ( 1 + ∑ k = 1 n μ ( k ) ⌊ n k ⌋ 2 ) = 3 π 2 n 2 + O ( n ( log  n ) 2 3 ( log  log  n ) 4 3 ) {\displaystyle \sum _{k=1}^{n}\varphi (k)={\tfrac {1}{2}}\left(1+\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor ^{2}\right)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)} ([22] cited in[23])
∑ k = 1 n φ ( k ) k = ∑ k = 1 n μ ( k ) k ⌊ n k ⌋ = 6 π 2 n + O ( ( log  n ) 2 3 ( log  log  n ) 4 3 ) {\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}=\sum _{k=1}^{n}{\frac {\mu (k)}{k}}\left\lfloor {\frac {n}{k}}\right\rfloor ={\frac {6}{\pi ^{2}}}n+O\left((\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)} [22]
∑ k = 1 n k φ ( k ) = 315 ζ ( 3 ) 2 π 4 n − log  n 2 + O ( ( log  n ) 2 3 ) {\displaystyle \sum _{k=1}^{n}{\frac {k}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}n-{\frac {\log n}{2}}+O\left((\log n)^{\frac {2}{3}}\right)} [24]
∑ k = 1 n 1 φ ( k ) = 315 ζ ( 3 ) 2 π 4 ( log  n + γ − ∑ p prime log  p p 2 − p + 1 ) + O ( ( log  n ) 2 3 n ) {\displaystyle \sum _{k=1}^{n}{\frac {1}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}\left(\log n+\gamma -\sum _{p{\text{ prime}}}{\frac {\log p}{p^{2}-p+1}}\right)+O\left({\frac {(\log n)^{\frac {2}{3}}}{n}}\right)} [24]
(where γ is the Euler–Mascheroni constant).
∑ gcd  ( k , m ) = 1 1 ≤ k ≤ n 1 = n φ ( m ) m + O ( 2 ω ( m ) ) {\displaystyle \sum _{\stackrel {1\leq k\leq n}{\operatorname {gcd} (k,m)=1}}\!\!\!\!1=n{\frac {\varphi (m)}{m}}+O\left(2^{\omega (m)}\right)}
where m > 1 is a positive integer and ω(m) is the number of distinct prime factors of m.[25]
Menon's identity[edit]
Main article: Menon's identity
In 1965 P. Kesava Menon proved
∑ gcd ( k , n ) = 1 1 ≤ k ≤ n gcd ( k − 1 , n ) = φ ( n ) d ( n ) , {\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\!\!\!\!\gcd(k-1,n)=\varphi (n)d(n),}
where d(n) = σ0(n) is the number of divisors of n.
Formulae involving the golden ratio[edit]
Schneider[26] found a pair of identities connecting the totient function, the golden ratio and the Möbius function μ(n). In this section φ(n) is the totient function, and ϕ = 1 + √5/2 = 1.618… is the golden ratio.
They are:
ϕ = − ∑ k = 1 ∞ φ ( k ) k log  ( 1 − 1 ϕ k ) {\displaystyle \phi =-\sum _{k=1}^{\infty }{\frac {\varphi (k)}{k}}\log \left(1-{\frac {1}{\phi ^{k}}}\right)}
and
1 ϕ = − ∑ k = 1 ∞ μ ( k ) k log  ( 1 − 1 ϕ k ) . {\displaystyle {\frac {1}{\phi }}=-\sum _{k=1}^{\infty }{\frac {\mu (k)}{k}}\log \left(1-{\frac {1}{\phi ^{k}}}\right).}
Subtracting them gives
∑ k = 1 ∞ μ ( k ) − φ ( k ) k log  ( 1 − 1 ϕ k ) = 1. {\displaystyle \sum _{k=1}^{\infty }{\frac {\mu (k)-\varphi (k)}{k}}\log \left(1-{\frac {1}{\phi ^{k}}}\right)=1.}
Applying the exponential function to both sides of the preceding identity yields an infinite product formula for e:
e = ∏ k = 1 ∞ ( 1 − 1 ϕ k ) μ ( k ) − φ ( k ) k . {\displaystyle e=\prod _{k=1}^{\infty }\left(1-{\frac {1}{\phi ^{k}}}\right)^{\frac {\mu (k)-\varphi (k)}{k}}.}
The proof is based on the two formulae
∑ k = 1 ∞ φ ( k ) k ( − log  ( 1 − x k ) ) = x 1 − x and ∑ k = 1 ∞ μ ( k ) k ( − log  ( 1 − x k ) ) = x , for 0 < x < 1. {\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }{\frac {\varphi (k)}{k}}\left(-\log \left(1-x^{k}\right)\right)&={\frac {x}{1-x}}\\{\text{and}}\;\sum _{k=1}^{\infty }{\frac {\mu (k)}{k}}\left(-\log \left(1-x^{k}\right)\right)&=x,\qquad \quad {\text{for }}0<x<1.\end{aligned}}}
Generating functions[edit]
The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as:[27]
∑ n = 1 ∞ φ ( n ) n s = ζ ( s − 1 ) ζ ( s ) . {\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}.}
The Lambert series generating function is[28]
∑ n = 1 ∞ φ ( n ) q n 1 − q n = q ( 1 − q ) 2 {\displaystyle \sum _{n=1}^{\infty }{\frac {\varphi (n)q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}}
which converges for |q| < 1.
Both of these are proved by elementary series manipulations and the formulae for φ(n).
Growth rate[edit]
In the words of Hardy & Wright, the order of φ(n) is “always ‘nearly n’.”[29]
First[30]
lim sup φ ( n ) n = 1 , {\displaystyle \lim \sup {\frac {\varphi (n)}{n}}=1,}
but as n goes to infinity,[31] for all δ > 0
φ ( n ) n 1 − δ → ∞ . {\displaystyle {\frac {\varphi (n)}{n^{1-\delta }}}\rightarrow \infty .}
These two formulae can be proved by using little more than the formulae for φ(n) and the divisor sum function σ(n).
In fact, during the proof of the second formula, the inequality
6 π 2 < φ ( n ) σ ( n ) n 2 < 1 , {\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\varphi (n)\sigma (n)}{n^{2}}}<1,}
true for n > 1, is proved.
We also have[20]
lim inf φ ( n ) n log  log  n = e − γ . {\displaystyle \lim \inf {\frac {\varphi (n)}{n}}\log \log n=e^{-\gamma }.}
Here γ is Euler's constant, γ = 0.577215665..., so eγ = 1.7810724... and e−γ = 0.56145948....
Proving this does not quite require the prime number theorem.[32][33] Since log log (n) goes to infinity, this formula shows that
lim inf φ ( n ) n = 0. {\displaystyle \lim \inf {\frac {\varphi (n)}{n}}=0.}
In fact, more is true.[34][35][36]
φ ( n ) > n e γ log  log  n + 3 log  log  n for n > 2 {\displaystyle \varphi (n)>{\frac {n}{e^{\gamma }\;\log \log n+{\frac {3}{\log \log n}}}}\quad {\text{for }}n>2}
and
φ ( n ) < n e γ log  log  n for infinitely many n . {\displaystyle \varphi (n)<{\frac {n}{e^{\gamma }\log \log n}}\quad {\text{for infinitely many }}n.}
The second inequality was shown by Jean-Louis Nicolas. Ribenboim says "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption."[36]
For the average order, we have[22][37]
φ ( 1 ) + φ ( 2 ) + ⋯ + φ ( n ) = 3 n 2 π 2 + O ( n ( log  n ) 2 3 ( log  log  n ) 4 3 ) as n → ∞ , {\displaystyle \varphi (1)+\varphi (2)+\cdots +\varphi (n)={\frac {3n^{2}}{\pi ^{2}}}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)\quad {\text{as }}n\rightarrow \infty ,}
due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N. M. Korobov (this is currently the best known estimate of this type). The "Big O" stands for a quantity that is bounded by a constant times the function of n inside the parentheses (which is small compared to n2).
This result can be used to prove[38] that the probability of two randomly chosen numbers being relatively prime is 6/π2.
Ratio of consecutive values[edit]
In 1950 Somayajulu proved[39][40]
lim inf φ ( n + 1 ) φ ( n ) = 0 and lim sup φ ( n + 1 ) φ ( n ) = ∞ . {\displaystyle {\begin{aligned}\lim \inf {\frac {\varphi (n+1)}{\varphi (n)}}&=0\quad {\text{and}}\\[5px]\lim \sup {\frac {\varphi (n+1)}{\varphi (n)}}&=\infty .\end{aligned}}}
In 1954 Schinzel and Sierpiński strengthened this, proving[39][40] that the set
{ φ ( n + 1 ) φ ( n ) , n = 1 , 2 , ⋯ } {\displaystyle \left\{{\frac {\varphi (n+1)}{\varphi (n)}},\;\;n=1,2,\cdots \right\}}
is dense in the positive real numbers. They also proved[39] that the set
{ φ ( n ) n , n = 1 , 2 , ⋯ } {\displaystyle \left\{{\frac {\varphi (n)}{n}},\;\;n=1,2,\cdots \right\}}
is dense in the interval (0,1).
Totient numbers[edit]
A totient number is a value of Euler's totient function: that is, an m for which there is at least one n for which φ(n) = m. The valency or multiplicity of a totient number m is the number of solutions to this equation.[41] A nontotient is a natural number which is not a totient number. Every odd integer exceeding 1 is trivially a nontotient. There are also infinitely many even nontotients,[42] and indeed every positive integer has a multiple which is an even nontotient.[43]
The number of totient numbers up to a given limit x is
x log  x e ( C + o ( 1 ) ) ( log  log  log  x ) 2 {\displaystyle {\frac {x}{\log x}}e^{{\big (}C+o(1){\big )}(\log \log \log x)^{2}}}
for a constant C = 0.8178146....[44]
If counted accordingly to multiplicity, the number of totient numbers up to a given limit x is
| { n : ϕ ( n ) ≤ x } | = ζ ( 2 ) ζ ( 3 ) ζ ( 6 ) ⋅ x + R ( x ) {\displaystyle {\Big \vert }\{n:\phi (n)\leq x\}{\Big \vert }={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}\cdot x+R(x)}
where the error term R is of order at most x/(log x)k for any positive k.[45]
It is known that the multiplicity of m exceeds mδ infinitely often for any δ < 0.55655.[46][47]
Ford's theorem[edit]
Ford (1999) proved that for every integer k ≥ 2 there is a totient number m of multiplicity k: that is, for which the equation φ(n) = m has exactly k solutions; this result had previously been conjectured by Wacław Sierpiński,[48] and it had been obtained as a consequence of Schinzel's hypothesis H.[44] Indeed, each multiplicity that occurs, does so infinitely often.[44][47]
However, no number m is known with multiplicity k = 1. Carmichael's totient function conjecture is the statement that there is no such m.[49]
Perfect totient numbers[edit]
Main article: Perfect totient number
Applications[edit]
Cyclotomy[edit]
Main article: Constructible polygon
In the last section of the Disquisitiones[50][51] Gauss proves[52] that a regular n-gon can be constructed with straightedge and compass if φ(n) is a power of 2. If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if n is a first power and n − 1 is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more.
Thus, a regular n-gon has a straightedge-and-compass construction if n is a product of distinct Fermat primes and any power of 2. The first few such n are[53]
2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... (sequence A003401 in the OEIS).
The RSA cryptosystem[edit]
Main article: RSA (algorithm)
Setting up an RSA system involves choosing large prime numbers p and q, computing n = pq and k = φ(n), and finding two numbers e and d such that ed ≡ 1 (mod k). The numbers n and e (the "encryption key") are released to the public, and d (the "decryption key") is kept private.
A message, represented by an integer m, where 0 < m < n, is encrypted by computing S = me (mod n).
It is decrypted by computing t = Sd (mod n). Euler's Theorem can be used to show that if 0 < t < n, then t = m.
The security of an RSA system would be compromised if the number n could be factored or if φ(n) could be computed without factoring n.
Unsolved problems[edit]
Lehmer's conjecture[edit]
Main article: Lehmer's totient problem
If p is prime, then φ(p) = p − 1. In 1932 D. H. Lehmer asked if there are any composite numbers n such that φ(n) | n − 1. None are known.[54]
In 1933 he proved that if any such n exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ω(n) ≥ 7). In 1980 Cohen and Hagis proved that n > 1020 and that ω(n) ≥ 14.[55] Further, Hagis showed that if 3 divides n then n > 101937042 and ω(n) ≥ 298848.[56][57]
Carmichael's conjecture[edit]
Main article: Carmichael's totient function conjecture
This states that there is no number n with the property that for all other numbers m, m ≠ n, φ(m) ≠ φ(n). See Ford's theorem above.
As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.[41]
See also[edit]
Carmichael function
Duffin–Schaeffer conjecture
Generalizations of Fermat's little theorem
Highly composite number
Multiplicative group of integers modulo n
Ramanujan sum
Notes[edit]
Jump up ^ "Euler's totient function". Khan Academy. Retrieved 2016-02-26.
Jump up ^ Long (1972, p. 85)
Jump up ^ Pettofrezzo & Byrkit (1970, p. 72)
Jump up ^ Long (1972, p. 162)
Jump up ^ Pettofrezzo & Byrkit (1970, p. 80)
Jump up ^ See Euler's theorem.
Jump up ^ L. Euler "Theoremata arithmetica nova methodo demonstrata" (An arithmetic theorem proved by a new method), Novi commentarii academiae scientiarum imperialis Petropolitanae (New Memoirs of the Saint-Petersburg Imperial Academy of Sciences), 8 (1763), 74–104. (The work was presented at the Saint-Petersburg Academy on October 15, 1759. A work with the same title was presented at the Berlin Academy on June 8, 1758). Available on-line in: Ferdinand Rudio, ed., Leonhardi Euleri Commentationes Arithmeticae, volume 1, in: Leonhardi Euleri Opera Omnia, series 1, volume 2 (Leipzig, Germany, B. G. Teubner, 1915), pages 531–555. On page 531, Euler defines n as the number of integers that are smaller than N and relatively prime to N (… aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, …), which is the phi function, φ(N).
^ Jump up to: a b Sandifer, p. 203
Jump up ^ Graham et al. p. 133 note 111
Jump up ^ L. Euler, Speculationes circa quasdam insignes proprietates numerorum, Acta Academiae Scientarum Imperialis Petropolitinae, vol. 4, (1784), pp. 18–30, or Opera Omnia, Series 1, volume 4, pp. 105–115. (The work was presented at the Saint-Petersburg Academy on October 9, 1775).
Jump up ^ Both φ(n) and ϕ(n) are seen in the literature. These are two forms of the lower-case Greek letter phi.
Jump up ^ Gauss, Disquisitiones Arithmeticae article 38
Jump up ^ J. J. Sylvester (1879) "On certain ternary cubic-form equations", American Journal of Mathematics, 2 : 357-393; Sylvester coins the term "totient" on page 361.
Jump up ^ "totient". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989.
Jump up ^ Schramm (2008)
Jump up ^ Gauss, DA, art 39
Jump up ^ Gauss, DA art. 39, arts. 52-54
Jump up ^ Graham et al. pp. 134-135
Jump up ^ The cell for n = 0 in the upper-left corner of the table is empty, as the function φ(n) is commonly defined only for positive integers, so it is not defined for n = 0.
^ Jump up to: a b Hardy & Wright 1979, thm. 328
Jump up ^ Dineva (in external refs), prop. 1
^ Jump up to: a b c Walfisz, Arnold (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte (in German). 16. Berlin: VEB Deutscher Verlag der Wissenschaften. Zbl 0146.06003.
Jump up ^ Lomadse, G., "The scientific work of Arnold Walfisz" (PDF), Acta Arithmetica, 10 (3): 227–237
^ Jump up to: a b Sitaramachandrarao, R. (1985). "On an error term of Landau II". Rocky Mountain J. Math. 15: 579–588.
Jump up ^ Bordellès in the external links
Jump up ^ All formulae in the section are from Schneider (in the external links)
Jump up ^ Hardy & Wright 1979, thm. 288
Jump up ^ Hardy & Wright 1979, thm. 309
Jump up ^ Hardy & Wright 1979, intro to § 18.4
Jump up ^ Hardy & Wright 1979, thm. 326
Jump up ^ Hardy & Wright 1979, thm. 327
Jump up ^ In fact Chebyshev's theorem (Hardy & Wright 1979, thm.7) and Mertens' third theorem is all that is needed.
Jump up ^ Hardy & Wright 1979, thm. 436
Jump up ^ Theorem 15 of Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers". Illinois J. Math. 6 (1): 64–94.
Jump up ^ Bach & Shallit, thm. 8.8.7
^ Jump up to: a b Ribenboim. The Book of Prime Number Records. Section 4.I.C. [full citation needed]
Jump up ^ Sándor, Mitrinović & Crstici (2006) pp.24–25
Jump up ^ Hardy & Wright 1979, thm. 332
^ Jump up to: a b c Ribenboim, p.38
^ Jump up to: a b Sándor, Mitrinović & Crstici (2006) p.16
^ Jump up to: a b Guy (2004) p.144
Jump up ^ Sándor & Crstici (2004) p.230
Jump up ^ Zhang, Mingzhi (1993). "On nontotients". Journal of Number Theory. 43 (2): 168–172. doi:10.1006/jnth.1993.1014. ISSN 0022-314X. Zbl 0772.11001.
^ Jump up to: a b c Ford, Kevin (1998). "The distribution of totients". Ramanujan J. 2 (1–2): 67–151. doi:10.1007/978-1-4757-4507-8_8. ISSN 1382-4090. Zbl 0914.11053.
Jump up ^ Sándor et al (2006) p.22
Jump up ^ Sándor et al (2006) p.21
^ Jump up to: a b Guy (2004) p.145
Jump up ^ Sándor & Crstici (2004) p.229
Jump up ^ Sándor & Crstici (2004) p.228
Jump up ^ Gauss, DA. The 7th § is arts. 336–366
Jump up ^ Gauss proved if n satisfies certain conditions then the n-gon can be constructed. In 1837 Pierre Wantzel proved the converse, if the n-gon is constructible, then n must satisfy Gauss's conditions
Jump up ^ Gauss, DA, art 366
Jump up ^ Gauss, DA, art. 366. This list is the last sentence in the Disquisitiones
Jump up ^ Ribenboim, pp. 36–37.
Jump up ^ Cohen, Graeme L.; Hagis, Peter, Jr. (1980). "On the number of prime factors of n if φ(n) divides n − 1". Nieuw Arch. Wiskd., III. Ser. 28: 177–185. ISSN 0028-9825. Zbl 0436.10002.
Jump up ^ Hagis, Peter, Jr. (1988). "On the equation M·φ(n) = n − 1". Nieuw Arch. Wiskd., IV. Ser. 6 (3): 255–261. ISSN 0028-9825. Zbl 0668.10006.
Jump up ^ Guy (2004) p.142
References[edit]
The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of Gauss' papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
References to the Disquisitiones are of the form Gauss, DA, art. nnn.
Abramowitz, M.; Stegun, I. A. (1964), Handbook of Mathematical Functions, New York: Dover Publications, ISBN 0-486-61272-4 . See paragraph 24.3.2.
Bach, Eric; Shallit, Jeffrey (1996), Algorithmic Number Theory (Vol I: Efficient Algorithms), MIT Press Series in the Foundations of Computing, Cambridge, MA: The MIT Press, ISBN 0-262-02405-5, Zbl 0873.11070
Ford, Kevin (1999), "The number of solutions of φ(x) = m", Annals of Mathematics, 150 (1): 283–311, doi:10.2307/121103, ISSN 0003-486X, JSTOR 121103, MR 1715326, Zbl 0978.11053 .
Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithmeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9
Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8
Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994), Concrete Mathematics: a foundation for computer science (2nd ed.), Reading, MA: Addison-Wesley, ISBN 0-201-55802-5, Zbl 0836.00001
Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics (3rd ed.), New York, NY: Springer-Verlag, ISBN 0-387-20860-7, Zbl 1058.11001
Hardy, G. H.; Wright, E. M. (1979), An Introduction to the Theory of Numbers (Fifth ed.), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
Ribenboim, Paulo (1996), The New Book of Prime Number Records (3rd ed.), New York: Springer, ISBN 0-387-94457-5, Zbl 0856.11001
Sandifer, Charles (2007), The early mathematics of Leonhard Euler, MAA, ISBN 0-88385-559-3
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006), Handbook of number theory I, Dordrecht: Springer-Verlag, pp. 9–36, ISBN 1-4020-4215-9, Zbl 1151.11300
Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 179–327. ISBN 1-4020-2546-7. Zbl 1079.11001.
Schramm, Wolfgang (2008), "The Fourier transform of functions of the greatest common divisor", Electronic Journal of Combinatorial Number Theory, A50 (8(1)) .
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Totient function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Euler's Phi Function and the Chinese Remainder Theorem — proof that φ(n) is multiplicative
Euler's totient function calculator in JavaScript — up to 20 digits
Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions
Plytage, Loomis, Polhill Summing Up The Euler Phi Function
[hide]
v
t
e
Totient function
Euler's totient function ϕ ( n ) {\displaystyle \phi (n)}
Jordan's totient function J k ( n ) {\displaystyle J_{k}(n)}
Carmichael function λ ( n ) {\displaystyle \lambda (n)}
Nontotient
Noncototient
Highly totient number
Highly cototient number
Sparsely totient number
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Open market operation
From Wikipedia, the free encyclopedia
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An open market operation (OMO) is an activity by a central bank to give (or take) liquidity in its currency to (or from) a bank or a group of banks. The central bank can either buy or sell government bonds in the open market (this is where the name was historically derived from) or, in what is now mostly the preferred solution, enter into a repo or secured lending transaction with a commercial bank: the central bank gives the money as a deposit for a defined period and synchronously takes an eligible asset as collateral. A central bank uses OMO as the primary means of implementing monetary policy. The usual aim of open market operations is—aside from supplying commercial banks with liquidity and sometimes taking surplus liquidity from commercial banks—to manipulate the short-term interest rate and the supply of base money in an economy, and thus indirectly control the total money supply, in effect expanding money or contracting the money supply. This involves meeting the demand of base money at the target interest rate by buying and selling government securities, or other financial instruments. Monetary targets, such as inflation, interest rates, or exchange rates, are used to guide this implementation.[1][2]
Contents
[hide]
1 Process of open market operations
2 Theoretical relationship to interest rates
3 Possible targets
4 How open market operations are conducted
4.1 United States
4.2 Eurozone
4.3 Switzerland
4.4 India
5 See also
6 References
7 External links
Process of open market operations[edit]
The central bank maintains loro accounts for a group of commercial banks, the so-called direct payment banks. A balance on such a loro account (it is a nostro account in the view of the commercial bank) represents central bank money in the regarded currency. Since central bank money currently exists mainly in the form of electronic records (electronic money) rather than in the form of paper or coins (physical money), open market operations can be conducted by simply increasing or decreasing (crediting or debiting) the amount of electronic money that a bank has in its reserve account at the central bank. This does not require the creation of new physical currency, unless a direct payment bank demands to exchange a part of its electronic money against banknotes or coins.
In most developed countries, central banks are not allowed to give loans without requiring suitable assets as collateral. Therefore, most central banks describe which assets are eligible for open market transactions. Technically, the central bank makes the loan and synchronously takes an equivalent amount of an eligible asset supplied by the borrowing commercial bank.
Theoretical relationship to interest rates[edit]
Classical economic theory postulates a distinctive relationship between the supply of central bank money and short-term interest rates: like for a commodity, a higher demand for central bank money would increase its price, the interest rate. When there is an increased demand for base money, the central bank must act if it wishes to maintain the short-term interest rate. It does this by increasing the supply of base money: it goes to the open market to buy a financial asset, such as government bonds. To pay for these assets, new central bank money is generated in the seller's loro account, increasing the total amount of base money in the economy. Conversely, if the central bank sells these assets in the open market, the base money is reduced.
Technically, the process works because the central bank has the authority to bring money in and out of existence. It is the only point in the whole system with the unlimited ability to produce money. Another organization may be able to influence the open market for a period of time, but the central bank will always be able to overpower their influence with an infinite supply of money.[3]
Side note: Countries that have a free floating currency not pegged to any commodity or other currency have a similar capacity to produce an unlimited amount of net financial assets (bonds), understandably, governments would like to utilize this capacity to meet other political ends like unemployment rate targeting, or relative size of various public services (military, education, health etc.), rather than any specific interest rate. Mostly, however the central bank is prevented by law or convention from giving way to such demands, being required to only generate central bank money in exchange for eligible assets (see above).
Possible targets[edit]
Under inflation targeting, open market operations target a specific short-term interest rate in the debt markets. This target is changed periodically to achieve and maintain an inflation rate within a target range. However, other variants of monetary policy also often target interest rates: the US Federal Reserve, the Bank of England and the European Central Bank use variations on interest rate targets to guide open market operations.
Besides interest rate targeting there are other possible targets of open markets operations. A second possible target is the contraction of the money supply, as was the case in the U.S. in the late 1970s through the early 1980s under Fed Chairman Paul Volcker.
Under a currency board open market operations would be used to achieve and maintain a fixed exchange rate with relation to some foreign currency.
Under a gold standard, notes would be convertible to gold, and so open market operations could be used to keep the value of a fiat currency constant relative to gold.
A central bank can also use a mixture of policy settings that change depending on circumstances. A central bank may peg its exchange rate (like a currency board) with different levels or forms of commitment. The looser the exchange rate peg, the more latitude the central bank has to target other variables (such as interest rates). It may instead target a basket of foreign currencies rather than a single currency. In some instances it is empowered to use additional means other than open market operations, such as changes in reserve requirements or capital controls, to achieve monetary outcomes.
How open market operations are conducted[edit]
United States[edit]
Further information: Monetary policy of the United States
In the United States, as of 2006[citation needed], the Federal Reserve sets an interest rate target for the Federal funds (overnight bank reserves) market. When the actual Federal funds rate is higher than the target, the New York Reserve Bank will usually increase the money supply via a repo (effectively borrowing from the dealers' perspective; lending for the Reserve Bank). When the actual Federal funds rate is less than the target, the Bank will usually decrease the money supply via a reverse repo (effectively lending from the dealers' perspective; borrowing for the Reserve Bank).
In the U.S., the Federal Reserve most commonly uses overnight repurchase agreements (repos) to temporarily create money, or reverse repos to temporarily destroy money, which offset temporary changes in the level of bank reserves.[4] The Federal Reserve also makes outright purchases and sales of securities through the System Open Market Account (SOMA) with its manager over the Trading Desk at the New York Reserve Bank. The trade of securities in the SOMA changes the balance of bank reserves, which also affects short-term interest rates. The SOMA manager is responsible for trades that result in a short-term interest rate near the target rate set by the Federal Open Market Committee (FOMC), or create money by the outright purchase of securities.[5] More rarely will it permanently destroy money by the outright sale of securities. These trades are made with a group of about 22 banks and bond dealers called primary dealers.
Money is created or destroyed by changing the reserve account of the bank with the Federal Reserve. The Federal Reserve has conducted open market operations in this manner since the 1920s, through the Open Market Desk at the Federal Reserve Bank of New York, under the direction of the Federal Open Market Committee. The open market operation is also a means through which inflation can be controlled because when treasury bills are sold to commercial banks these banks can no longer give out loans to the public for the period and therefore money is being reduced from circulation.
Eurozone[edit]
The European Central Bank has similar mechanisms for their operations; it describes its methods as a four-tiered approach with different goals: beside its main goal of steering and smoothing Eurozone interest rates while managing the liquidity situation in the market the ECB also has the aim of signalling the stance of monetary policy with its operations.
Broadly speaking, the ECB controls liquidity in the banking system via Refinancing Operations, which are basically repurchase agreements,[6] i.e. banks put up acceptable collateral with the ECB and receive a cash loan in return. These are the following main categories of refinancing operations that can be employed depending on the desired outcome:
The regular weekly main refinancing operations (MRO) with maturity of one week and,
The monthly longer-term refinancing operations (LTRO) provide liquidity to the financial sector, while ad hoc
"Fine-tuning operations" (in the form of reverse or outright transactions, foreign exchange swaps and the collection of fixed-term deposits) aim to smooth interest rates caused by liquidity fluctuations in the market and
"Structural operations" are used to adjust the central banks' longer-term structural positions vis-a-vis the financial sector.
Refinancing operations are conducted via an auction mechanism. The ECB specifies the amount of liquidity it wishes to auction (called the allotted amount) and asks banks for expressions of interest. In a fixed rate tender the ECB also specifies the interest rate at which it is willing to lend money; alternatively, in a variable rate tender the interest rate is not specified and banks bid against each other (subject to a minimum bid rate specified by the ECB) to access the available liquidity.
MRO auctions are held on Mondays, with settlement (i.e. disbursal of the funds) occurring the following Wednesday. For example, at its auction on 6 October 2008, the ECB made available 250 million in EUR on 8 October at a minimum rate of 4.25%. It received 271 million in bids, and the allotted amount (250) was awarded at an average weighted rate of 4.99%.
Since mid-October 2008, however, the ECB has been following a different procedure on a temporary basis, the fixed rate MRO with full allottment. In this case the ECB specifies the rate but not the amount of credit made available, and banks can request as much as they wish (subject as always to being able to provide sufficient collateral). This procedure was made necessary by the financial crisis of 2008 and is expected to end at some time in the future.
Though the ECB's main refinancing operations (MRO) are from repo auctions with a (bi)weekly maturity and monthly maturation, Longer-Term Refinancing Operations (LTROs) are also issued, which traditionally mature after three months; since 2008, tenders are now offered for six months, 12 months and 36 months.[7]
Switzerland[edit]
The Swiss National Bank currently targets the 3-month Swiss franc LIBOR rate. The primary way the SNB influences the 3-month Swiss franc LIBOR rate is through open market operations, with the most important monetary policy instrument being repo transactions.[8]
India[edit]
India’s Open Market Operation is much influenced by the fact that it is a developing country and that the capital flows are very different from those in developed countries. Thus Reserve Bank of India (India's central bank) has to make policies and use instruments accordingly. Prior to the 1991 financial reforms, RBI’s major source of funding and control over credit and interest rates was the CRR (Cash reserve ratio) and the SLR (Statutory Liquidity Ratio). But after the reforms, the use of CRR as an effective tool was de-emphasized and the use of open market operations increased. OMO’s are more effective in adjusting [market liquidity].
The two traditional type of OMO’s used by RBI:
Outright purchase (PEMO): Is outright buying or selling of government securities. (Permanent).
Repurchase agreement (REPO): Is short term, and are subject to repurchase.[9]
However, even after sidelining CRR as an instrument, there was still less liquidity and skewedness in the market. And thus, on the recommendations of the Narsimham Committee Report (1998), The RBI brought together a Liquidity Adjustment Facility (LAF). It commenced in June, 2000, and it was set up to oversee liquidity on a daily basis and to monitor market interest rates. For the LAF, two rates are set by the RBI: repo rate and reverse repo rate. The repo rate is applicable while selling securities to RBI (daily injection of liquidity), while the reverse repo rate is applicable when banks buy back those securities (daily absorption of liquidity). Also, these interest rates fixed by the RBI also help in determining other market interest rates.[10]
India experiences large capital inflows every day, and even though the OMO and the LAF policies were able to withhold the inflows, another instrument was needed to keep the liquidity intact. Thus, on the recommendations of the Working Group of RBI on instruments of Sterilization (December, 2003), a new scheme known as the Market stabilization scheme (MSS) was set up. The LAF and the OMO’s were dealing with day-to-day liquidity management, whereas the MSS was set up to sterilize the liquidity absorption and make it more enduring.[11]
According to this scheme, the RBI issues additional T-bills and securities to absorb the liquidity. And the money goes into the Market Stabilization scheme Account (MSSA). The RBI cannot use this account for paying any interest or discounts and cannot credit any premiums to this account. The Government, in collaboration with the RBI, fixes a ceiling amount on the issue of these instruments.[12]
See also[edit]
Fiat currency
Fractional Reserve Banking
Interest rate
List of economics topics
List of finance topics
Modern Monetary Theory
Monetarism
Money creation
Quantitative easing
OMOs a tool for Inflation and Interest rate targeting
References[edit]
Jump up ^ Open market operations: A Glossary of Political Economy Terms - Dr. Paul M. Johnson
Jump up ^ Open Market Operations - William F. Hummel
Jump up ^ Jeff (2013-09-17). "Federal Reserve: How and Why Do They Change Interest Rates?". Trader Brains Blog. www.radbrains.com. Retrieved 2013-10-08.
Jump up ^ Repurchase and Reverse Repurchase Transactions - Fedpoints - Federal Reserve Bank of New York
Jump up ^ Open Market Operation - Fedpoints - Federal Reserve Bank of New York
Jump up ^ "European Central Bank". FXPedia. Retrieved 2011-09-19.
Jump up ^ "ECB offers longer-term finance via six-month LTROs". May 2008.
Jump up ^ "Monetary policy instruments (situation in 2009)". Swiss National Bank. Retrieved 1 March 2011.
Jump up ^ "Central Banks Guide".
Jump up ^ Economics by Paul Anthony Samuelson.
Jump up ^ "The Hindu: Features Of stabilization scheme".
Jump up ^ "Implementation Of Monetary Policy".
External links[edit]
Understanding Open market operation by M. A. Akhtar
Federal Reserve System OMC
European Central Bank OMC
Federal Reserve Bank of New York: Open Market Operations
CRR, SLR and the Repo Rate - Monetary Policy tools in India
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