A Riemann jarrabawa. Rarraba Firayim lambobin

A shekarar 1900, daya daga cikin mafi girma da masana kimiyya na karni na karshe, David Hilbert sanya jerin kunshi 23 ta kasa warwaruwa matsaloli na lissafi. Aiki a kan su ya na da wani gagarumin tasiri a kan ci gaban wannan filin na mutum ilmi. Bayan shekaru 100 a cikin Clay Ilmin Lissafi Cibiyar gabatar da wani jerin bakwai matsaloli, da aka sani da Millennium manufofin. Da hukuncin kowane daga cikinsu da aka miƙa kyauta daga $ 1 da miliyan.

Sai dai matsalar, wanda kuma yana daga cikin biyu lists na wasanin gwada ilimi, domin ƙarni bai ba sauran su masana kimiyya, ya zama Riemann jarrabawa. Ta aka har yanzu jiran ya yanke shawara.

Brief sada bayanai

Georg Friedrich Bernhard Riemann aka haife shi a 1826 a Hanover, a wani babban iyali na wani matalauci fasto, kuma ya rayu ne kawai shekaru 39 da haihuwa. Da ya gudanar ya buga 10 takardunku. Duk da haka, a lokacin rayuwa da Riemann ya dauke da wani magaji na malaminsa Johann Gauss. A shekaru 25 da matasa masanin kimiyya kare rubutun "Harsashen ka'idar ayyuka na wani hadadden m." Daga baya ya tsara jarrabawa, wanda ya zama sananne.

primes

LISSAFI zo a lokacin da mutum ya koya ƙidãya. Sa'an nan ya tashi na farko ra'ayin da lambobi, wanda daga baya kokarin rarraba. An lura da cewa wasu daga cikinsu suna da kowa Properties. A musamman, daga cikin halitta lambobin m. E. Wadanda wanda aka yi amfani da lissafi (lambobin) ko da sanya yawan abubuwa da aka kasaftawa wani rukuni na irin wanda aka raba kawai ta daya da kansu. Suka kira sauki. An m hujja daga cikin Theorem iyaka sa na lambobin da aka ba da Euclid a cikin "Abubuwa". A lokacin, mun kuma suna ci gaba da bincike. A musamman, da most of mai yawan san ² 74207281 - 1.

Euler ta dabara

Tare da mas'ala ta izuwa yawa primes Euclid tsare da na biyu Theorem kadai zai yiwu factorization. A cewar shi da wani m lamba ne samfurin na daya kawai sa na primes. A 1737, babban Jamus lissafi Leonhard Euler bayyana farko na Euclid ta Theorem a kan rashin iyaka da dabara aka nuna a kasa.

Shi ne ake kira da zeta aiki, inda s - akai da kuma p ne duk sauki dabi'u. Daga shi kai tsaye bi da kuma amincewa da bambancinsa na fadada daga Euclid.

Riemann zeta aiki

Euler ta dabara a kusa dubawa ne quite na ƙwarai, kamar yadda aka ba da rabo tsakanin sauki da kuma integers. Bayan duk, a cikin ta gefen hagu ana yawaita izuwa yawa maganganu cewa ya dogara ne kawai a kan sauki, kuma a cikin da hakkin adadin da ake dangantawa da duk m integers.

Riemann tafi a kan Euler. Domin samun da key da matsalar rarraba da lambobi, shi ne samarwa domin ayyana da dabara duka biyu na ainihi da kuma hadadden m. Ita ce ta wanda daga baya ya zama sananne a matsayin Riemann zeta aiki. A shekarar 1859, masana kimiyya da aka buga wata kasida mai suna "A cikin yawan primes cewa kada ku ƙẽtare qaddara darajar", wanda taƙaice duk su ideas.

Riemann samarwa da yin amfani da wani yawan Euler, convergent ga duk real s> 1. Idan wannan dabara da ake amfani da hadaddun s, sa'an nan cikin jerin zai converge ga wani darajar da m tare da real part ne mafi girma daga 1. Riemann amfani da nazari ci gaba da hanya ta fadada definition of zeta (s) ga dukan hadaddun lambobi, amma "amai" naúrar. Shi ba zai yiwu ba, domin idan s = 1 zeta aiki ƙaruwa to rashin iyaka.

m ji

The tambaya: abin da yake ban sha'awa da kuma muhimmanci zeta aiki, wanda yana da muhimmanci a cikin aiki na Riemann a kan null jarrabawa? Kamar yadda ka sani, a wannan lokacin ba a samu wani sauki, abin kõyi da ya bayyana da rarraba Firayim lambobi daga cikin halitta. Riemann iya gane cewa yawan pi (x) na Firayim lambobi, wanda suke ba fifiko ba ga x, an bayyana ta da rarraba nontrivial sifili zeta aiki. Bugu da ƙari, Riemann jarrabawa ne mai zama dole yanayin domin tabbatar da wucin gadi kimantawa na wasu Hikimar lissafi mai tsauri.

A Riemann jarrabawa

Daya daga cikin na farko formulations wannan ilmin lissafi matsala, ba tabbatar da wannan rana, shi ne: maras muhimmanci 0 zeta aiki - hadaddun lambobin da real part daidai ½. A wasu kalmomin, suna shirya a kan wani madaidaiciya line Re s = ½.

Akwai kuma wani jimlace Riemann jarrabawa, wanda shi ne guda bayani, amma domin hakan na zeta-ayyuka, wanda ake kira da Dirichlet (ga. Photo kasa) L-ayyuka.

A cikin dabara χ (n) - na lamba harafin (na zamani k).

Riemann ta sanarwa ne da ake kira null jarrabawa, kamar yadda aka tabbatar ga daidaito da data kasance samfuri bayanai.

Kamar yadda na yi jãyayya Riemann

Note Jamus lissafi aka asali tsara quite yo. Gaskiyar ita ce, a lokacin da masanin kimiyyar da aka je tabbatar da wani Theorem a kan rarraba Firayim lambobi, da kuma a cikin wannan mahallin, wannan jarrabawa ba shi da yawa sakamako. Duk da haka, da rawar a magance da yawa wasu al'amurran da suka shafi ne babban. Wannan shi ne dalilin da ya sa Riemann jarrabawa a yanzu masana kimiyya da dama gane muhimmanci na unproven ilmin lissafi matsaloli.

Kamar yadda aka ce, a tabbatar da Theorem a kan rarraba da cikakken Riemann jarrabawa ba lallai ba ne, da kuma quite Azancin tabbatar da cewa real wani ɓangare na wani maras maras muhimmanci sifili na zeta aiki ne tsakanin 0 kuma 1. Wannan dukiya ya nuna cewa jimlar duk 0-m zeta aiki da ya bayyana a cikin ainihin dabara sama, - guntun m. Ga manyan dabi'u na x, shi zai iya duka a rasa. The kawai memba na dabara, wanda zai zama canzawa ko a sosai high x, x ne da kansa. Sauran hadaddun sharuddan a kwatanta da shi asymptotically bace. Saboda haka, mai nauyi Naira Miliyan Xari o ƙarin x. Wannan hujja za a iya gani a matsayin hujja na gaskiya na Firayim yawan Theorem. Saboda haka, zeros na Riemann zeta aiki bayyana na musamman rawa. Shi ne ya tabbatar da cewa wadannan dabi'u ba zai iya taimaka muhimmanci ga fadada dabara.

Riemann mabiya

The ban tausayi mutuwa daga tarin fuka ya hana masanin kimiyya kawo ga ma'ana karshen shirin. Duk da haka, ya ɗauki baton daga W-F. de la Vallée Poussin da Zhak Adamar. Da kansa daga juna suka janye Firayim yawan Theorem. Hadamard da Poussin gudanar ya tabbatar da cewa duk nontrivial 0 zeta aiki suna located a cikin m band.

Godiya ga aikin wadannan masana kimiyyar, wani sabon reshe na lissafi - hikimar tantance ka'idar lambobi. Daga baya, wasu masu bincike sun samu kadan mafi m hujja daga cikin Theorem aka aiki a Roma. A musamman, Harbhajan Erdös da Atle Selberg sun bude ko ya tabbatar da rikitacciyar sarkar na dabaru, ba ya bukatar da yin amfani da hadaddun analysis. Duk da haka, a wannan lokaci da ra'ayin Riemann da dama muhimmanci theorems da aka tabbatar da, ciki har da kimantawa na da yawa ayyuka na yawan ka'idar. A dangane da wannan sabon aiki Erdős da Atle Selberg kusan wani abu ba ya canzawa.

Daya daga cikin sauki da kuma mafi kyau shaida daga cikin matsalar da aka samu a 1980 da Donald Newman. Yana da aka bisa ga sanannun Cauchy Theorem.

Barazana idan Riemann ta jarrabawa ne tushen zamani cryptography

Data boye-boye fito da bayyanar haruffa, ko kuma wajen, su da kansu za a iya daukarsa a matsayin na farko code. A lokacin, akwai dukan sabon Trend na dijital cryptography, wanda ke tsunduma a cikin ci gaban da boye-boye lissafi mai tsauri.

Simple da kuma "Semisimple" yawan m. E. Wadanda wanda ake kawai kasu kashi biyu wasu lambobi daga cikin wannan aji, su ne tushen jama'a key tsarin, da aka sani da RSA. Yana yana da fadi da aikace-aikace. A musamman, shi da ake amfani a cikin ƙarni na wani lantarki sa hannu. Idan muka magana cikin sharuddan da samuwa "teapot", da Riemann jarrabawa ta riki da wanzuwar tsarin a rarraba Firayim lambobi. Saboda haka, rage muhimmanci juriya na Hikimar keys, a kan wanda ya dogara da aminci online ma'amaloli a e-kasuwanci.

Sauran ta kasa warwaruwa ilmin lissafi matsaloli

Complete labarin da daraja duqufad da 'yan kalmomi zuwa wasu ayyuka na Millennium. Wadannan sun hada da:

• Daidaitan azuzuwan P da kuma NP. Matsalar da aka tsara kamar haka: idan wani tabbatacce amsar wani ba tambaya ne tabbatar a polynomial lokaci, sa'an nan ne gaskiya cewa shi da kansa amsar wannan tambaya za a iya samu da sauri?
• Hodge zato. A sauki sharuddan da shi za a iya bayyana kamar haka: ga wasu iri projective algebraic manifolds (sarari) Hodge hawan keke ne haduwa da abubuwa da suke da wani lissafi da fassarar, watau algebraic hawan keke ...
• Poincaré zato. Shi ne kawai tabbatar a lokacin Millennium matsaloli. A cewar shi da wani uku-girma abu da ciwon musamman Properties na 3-girma Sphere, Sphere dole ne daidai to nakasawa.
• Yabo na jimla Yang - Mills ka'idar. Muna da bukatar su tabbatar da cewa jimla ka'idar, sa a gaba ta hanyar wadannan masana da sarari R ^4, akwai wani 0-taro lahani ga wani sauki k daga wani m kungiyar G.
• A cikin jarrabawa na Birch - Swinnerton-Dyer. Wannan shi ne wani matsala cewa shi ne dacewa da cryptography. Yana shafi elliptical masu lankwasa.
• Matsalar da wanzuwar kuma smoothness na mafita daga cikin Navier - dana kunamar rura lissafai.

Yanzu ka san Riemann jarrabawa. A sauki sharuddan, mun tsara da kuma wasu daga cikin sauran manufofin da karni. Gaskiyar cewa za su iya warware ko an tabbatar da cewa ba su da wani bayani - yana da wani al'amari na lokacin. Kuma wannan shi ne kamar wuya ya yi jira tsayi da yawa, kamar yadda da lissafi suna ƙara yin amfani da mai aiki da na'urar kwamfuta ikon kwakwalwa. Duk da haka, ba kome ne batun da art da kuma warware matsalolin kimiyya da farko bukatar diraya da kerawa.