Función de Liouville

La función de Liouville, denotada por λ(n) y atribuida a Joseph Liouville, es una importante función en teoría de números.

Si n es un entero positivo, entonces λ(n) es definido como:

\lambda (n)=(-1)^{\Omega (n)},\,\!

donde la función Ω(n) es el número de factores primos de n, contados con multiplicidad. Véase la (sucesión A008836 en OEIS).

λ es una función completamente multiplicativa dado que Ω(n) es una función aditiva. Debido a que Ω(1) = 0 tenemos que λ(1) = 1. La función de Liouville satisface la siguiente identidad:

\sum _{d|n}\lambda (d)=1\,\!

si n es un cuadrado perfecto, y:

\sum _{d|n}\lambda (d)=0\,\!

de otro modo.

Series

La serie de Dirichlet para la función de Liouville en términos de la función zeta de Riemann tiene la forma

{\frac {\zeta (2s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}.

La serie de Lambert para la función de Liouville es

\sum _{n=1}^{\infty }{\frac {\lambda (n)q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}={\frac {1}{2}}\left(\vartheta _{3}(q)-1\right),

donde $\vartheta _{3}(q)$ es la función theta de Jacobi.

Conjeturas

La conjetura de Pólya es una conjetura formulada por George Pólya en 1919, esta establece que:

L(n)=\sum _{k=1}^{n}\lambda (k)\leq 0

para n > 1. Esta conjetura resultó ser falsa. El contraejemplo más pequeño es n = 906150257, encontrado por Minoru Tanaka en 1980. Se desconoce si L(n) cambia de signo infinitamente.

Definiendo la suma

M(n)=\sum _{k=1}^{n}{\frac {\lambda (k)}{k}},

se especula que en algún momento M(n) ≥ 0 para n ≥ n₀ suficientemente grande (esta conjetura es ocasionalmente -pero incorrectamente- atribuida a Pál Turán). Esta fue refutada por Haselgrove en 1958 (véase la referencia), él mostró que M(n) toma valores negativos infinitas veces. De ser verdad esta conjetura, esta se puede ver como una prueba de la hipótesis de Riemann, como lo mostró Pál Turán.

Referencias

Polya, G., Verschiedene Bemerkungen zur Zahlentheorie. Jahresbericht der deutschen Math.-Vereinigung 28 (1919), 31–40.
Haselgrove, C.B. A disproof of a conjecture of Polya. Mathematika 5 (1958), 141–145.
Lehman, R., On Liouville's function. Math. Comp. 14 (1960), 311–320.
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function. Tokyo Journal of Mathematics 3, 187–189, (1980).

Datos: Q385019
Multimedia: Liouville function / Q385019

Read other articles:

Parlament

Lagstiftande församlingKammareEnkammarsystem · Tvåkammarsystem Trekammarsystem · Flerkammarsystem Överhus (Senat) · UnderhusParlamentParlamentarism · Parlamentsgrupp Parlamentsledamot · Internationella parlamentParlamentariskt styrelsesättKommitté · Kvorum · Motion (Misstroendeförklaring)TyperKongress (Kongressledamot) · Kommunestyre (Rådman) · Ståndriksdag (Stånd)Lagstiftande församlingar i erkända stater För det sven...

فلسفة علم الأحياء

تهتم فلسفة علم الأحياء بدراسة العلاقة بين الأفكار والفروض العلمية التي يتقدم بها العلماء والفلاسفة ومن أقدم من أهتم بمثل هذه الدراسات الفلاسفة اليونان القدماء مثل : طاليس وأناكسيمانس وأكسينوفان والمدرسة الفلسفية الأيونية بشكل محدد وكذلك المدرسة الآيلية في مسألة نشأة...

Первородный грех

Об экономическом термине см. Первородный грех (экономика). ХристианствоБиблия Ветхий Завет Новый Завет Евангелие Десять заповедей Нагорная проповедь Апокрифы Бог, Троица Бог Отец Иисус Христос Святой Дух История христианства Апостолы Хронология христианства Ран�...

Secretary's Distinguished Service Award

Secretary's Distinguished Service AwardTypeMedalAwarded forRecognition of exceptionally outstanding leadership, professional competence, and significant accomplishment over a sustained period of time in the field of foreign affairsPresented byUnited States Department of StateEligibilityForeign Service, Civil Service, U.S. military, civiliansStatusCurrently awardedRibbon PrecedenceNext (lower)Secretary’s Award The Secretary's Distinguished Service Award is an award of the United States...

Kunsthalle Emden

Kunsthalle Emden Kunsthalle Emden (Henri and Eske Nannen Foundation, Donation Otto Van de loo) is a German art museum in Emden in East Frisia, Germany.[1][2] The museum's collections include more than 1,500 works.[3] The Kunsthalle in December 2007. The Kunsthalle in April 2008. History In 1986, Henri Nannen (1913–1996) commissioned a building for his collection of 20th-century art in Emden, his home town. The core of the collection includes German Expressionism with...

Élections législatives de 1962 dans l'Yonne

1958 1967 Élections législatives de 1962 dans l'Yonne 3 sièges de députés à l'Assemblée nationale 18 et 25 novembre 1962 Corps électoral et résultats Inscrits 170 101 Votants au 1er tour 113 921 66,97 % 8,4 Votes exprimés au 1er tour 108 745 Votants au 2d tour 81 334 69,91 % Votes exprimés au 2d tour 77 152 Majorité présidentielle Liste Union pour la nouvelle République (UDT)Républicains indépendantsModérés Voix au ...

90

89 ← 90 → 91素因数分解 2 × 32 × 5二進法 1011010三進法 10100四進法 1122五進法 330六進法 230七進法 156八進法 132十二進法 76十六進法 5A二十進法 4A二十四進法 3I三十六進法 2Iローマ数字 XC漢数字九十大字九拾算木 90（九十、きゅうじゅう、ここのそ、ここそじ）は自然数、また整数において、89の次で91の前の数である。性質 90 は合成数であり、約数は 1, 2, 3, 5, 6, 9, 10, ...

Service-oriented architecture

Architectural pattern in software design In software engineering, service-oriented architecture (SOA) is an architectural style that focuses on discrete services instead of a monolithic design.[1] By consequence, it is also applied in the field of software design where services are provided to the other components by application components, through a communication protocol over a network. A service is a discrete unit of functionality that can be accessed remotely and acted upon and up...

Multan

Multan مُلتان Mausolée de Shah Rukn-e-Alam à Multan. Administration Pays Pakistan Province Pendjab Division Multan District Multan Maire Mian Faisal Mukhtar Démographie Population 1 871 843 hab. (rec. 2017[1]) Densité 2 397 hab./km2 Géographie Coordonnées 30° 12′ nord, 71° 25′ est Altitude 122 m Superficie 78 100 ha = 781 km2 Localisation Géolocalisation sur la carte : Pakistan Multan Géolocal...

Robert Latou Dickinson

American obstetrician This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Robert Latou Dickinson – news · newspapers · books · scholar · JSTOR (February 2013) (Learn how and when to remove this message) Robert L. DickinsonBornRobert Latou Dickinson(1861-02-21)February 21, 1861Jersey City, New Jersey, USDiedNovem...

2007 German motorcycle Grand Prix

Tenth round of the 2007 MotoGP championship 2007 German Grand PrixRace detailsRace 10 of 18 races in the2007 Grand Prix motorcycle racing seasonDate15 July 2007Official nameAlice Motorrad Grand Prix Deutschland[1][2][3]LocationSachsenringCoursePermanent racing facility3.671 km (2.281 mi)MotoGPPole positionRider Casey StonerTime 1:22.384Fastest lapRider Dani PedrosaTime 1:23.082PodiumFirst Dani PedrosaSecond Loris CapirossiThird Nicky Hayden250 ccP...

Semiasse maggiore

Questa voce o sezione sull'argomento geometria non cita le fonti necessarie o quelle presenti sono insufficienti. Puoi migliorare questa voce aggiungendo citazioni da fonti attendibili secondo le linee guida sull'uso delle fonti. Segui i suggerimenti del progetto di riferimento. In geometria, il semiasse maggiore a {\displaystyle a} è un concetto geometrico applicabile all'ellisse e all'iperbole. Indice 1 Ellisse 2 Iperbole 3 Astronomia 4 Collegamenti esterni Ellisse a è il semiasse m...

Electron degeneracy pressure

Repulsive force in quantum mechanics In astrophysics and condensed matter physics, electron degeneracy pressure is a quantum mechanical effect critical to understanding the stability of white dwarf stars and metal solids. It is a manifestation of the more general phenomenon of quantum degeneracy pressure. In metals and white dwarf stars, electrons can be modeled as a gas of non-interacting electrons confined to a finite volume. In reality, there are strong electromagnetic forces between the n...

Ryan Sweeting

American tennis player Ryan SweetingSweeting at the 2009 US OpenCountry (sports) United StatesResidenceFort Lauderdale, Florida, United StatesBorn (1987-07-14) July 14, 1987 (age 36)Nassau, BahamasHeight1.96 m (6 ft 5 in)Turned pro2007Retired2015PlaysRight-handed (two-handed backhand)Prize money$1,024,486SinglesCareer record35–57 (38.0%)Career titles1Highest rankingNo. 64 (12 September 2011)Grand Slam singles resultsAustralian Open2R (2...

The Cleveland Gazette

Weekly newspaper (1883-1945) Harry C. Smith, founder, editor and proprietor of The Cleveland Gazette for much of its early existence The Cleveland Gazette was a weekly newspaper published in Cleveland from August 25, 1883, to May 20, 1945. It was an African American newspaper owned and edited by Harry Clay Smith, initially with a group of partners. Circulation was estimated between 5,000 and 18,000. The Gazette became the longest-publishing African American weekly in the United States, earnin...

Black Chamber

American cryptanalytic organization For the postal monitoring office, see Cabinet noir. For the song by Blind Guardian, see Somewhere Far Beyond. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Black Chamber – news · newspapers · books · scholar · JSTOR (May 2011) (Learn how and when to remove this message) ...

Georges Lemaître

Belgian scientist and Catholic priest (1894–1966) This article is about the Belgian physicist and priest. For the spacecraft, see Georges Lemaître ATV. For the American physician, see George D. LeMaitre. For the professional road bicycle racer, see Georges Lemaire. The Reverend MonsignorGeorges LemaîtreRAS AssociateLemaître in 1933BornGeorges Henri Joseph Édouard Lemaître(1894-07-17)17 July 1894Charleroi, BelgiumDied20 June 1966(1966-06-20) (aged 71)Leuven, BelgiumAlma materC...

Revermont

Pour les articles homonymes, voir Val-Revermont. Revermont Paysage typique du Revermont à Corveissiat. Subdivision administrative Auvergne-Rhône-Alpes, Bourgogne-Franche-Comté Subdivision administrative Ain et Jura Villes principales Val-Revermont et Ceyzériat Coordonnées 46° 10′ 40″ nord, 5° 20′ 30″ est Superficie approximative 500 km2 Géologie Région karstique propice aux grottes Relief Relief montagneux du pré-Jura Production Agricultur...

Selección de voleibol de Yugoslavia

Selección de voleibol de Yugoslavia Archivo:Flag of Yugoslavia (1943-1992).svgDatos generalesPaís YugoslaviaConfederación CEVRanking FIVB Campeonato MundialParticipaciones 4 (primera vez en 1956)Mejor resultado 8.º puesto en 1962 y 1966Campeonato EuropeoParticipaciones 13 (primera vez en 1951)Mejor resultado en 1975 y 1979Juegos OlímpicosParticipaciones 1 (primera vez en 1980)Mejor resultado 6.º puesto en 1980[editar datos en Wikidata] La Selección de voleibol de Yugoslavia f...

ダチョウ目

ダチョウ目生息年代: 暁新世-完新世 56–0 Ma PreЄ Є O S D C P T J K Pg N ダチョウ分類ドメイン : 真核生物 Eukaryota 界 : 動物界 Animalia 門 : 脊索動物門 Chordata 亜門 : 脊椎動物亜門 Vertebrata 綱 : 鳥綱 Aves 上目 : 古顎上目 Palaeognathae 目 : ダチョウ目 Struthioniformes 学名 Struthioniformes Latham, 1790 ダチョウ目（ダチョウもく、学名：Struthioniformes）は鳥類の1目である。ダチョウ科（...