Te Rēimāna whakapae. Tohatoha o tau pirimia

I roto i te 1900, tetahi o nga pūtaiao nui o te rau tau whakamutunga, e Rawiri Hilbert hanga he rārangi tënei o 23 ngā raru kāore o pāngarau. i te mahi i runga i a ratou kua he pānga nui i runga i te whanaketanga o tenei mara o te matauranga tangata. I muri 100 tau i roto i te Clay Mathematical Institute aroaro he rārangi o ngā raruraru e whitu, e mohiotia ana ko te whāinga Mileniuma. Hoki te whakatau o ia o ratou i patua ana te utu o $ 1 miriona.

Te raruraru anake, i i roto i te rārangi e rua o panga, kihai hoki tenetere i hoatu okiokinga ki pūtaiao, ka te whakapae Rēimāna. Kei te tatari tonu ia mo tona whakatau.

mōhiohio haurongo Poto

whanau Georg Friedrich Peikari Rēimāna i roto i 1826 i roto i Hanover, i roto i te utuafare rahi o te hepara rawakore, a noho ana tau 39 anake tau. whakahaere ia ki te whakaputa i 10 pepa. Heoi, i roto i te ora o Rēimāna whakaaro ia he whakakapi o tona kaiwhakaako Johann Gauss. I 25 tau mau pūtaiao taitamariki tona tuhinga "putake o te ariā o ngā mahi o te tāupe matatini." I muri whakatakoto ia tona whakapae, i meinga rongonui.

primes

haere mai Pāngarau ka ako te tangata ki te tatau. Katahi ka whakatika te whakaaro tuatahi o nga tau, i muri tamata ki whakarōpū. Ua itehia e, e whai etahi o ratou āhuatanga noa. I roto i ngā, i roto i nga tau tūturu m. E. hunga i whakamahia nei i roto i te tātai (tauanga) te maha whakaritea o ngā tūemi ranei kua tohaina he rōpū o taua e kua wehea anake e kotahi me ratou. i huaina ratou ohie. He tohu huatau o te ture tauwehe huinga mure ore o tau i homai e Euclid i tona "Elements". I te kau, e tonu tatou i to ratou rapu. I roto i ngā, i te nui rawa o te maha o mōhiotia 2 ^74207281 - 1.

tātai o Hēura

Āpiti atu ki te ariā o primes ore maha Euclid tautuhi me te tuarua o nga ture tauwehe te factorization taea anake. E ai ki reira tetahi tau tōpū pai ko te hua o tetahi anake huinga o primes. I roto i 1737, te mathematician nui German Leonhard Hēura faaite tuatahi o te ture tauwehe o Euclid i te mutu- o te tātai e whakaaturia i raro nei.

Kei te karanga a te reira i te mahi Zeta, i reira s - he tamau me te p, ko te uara ohie katoa. Mai i reira tika aru me te whakaaetanga o te mana āhua o te roha o Euclid.

Rēimāna mahi Zeta

tātai o Hēura i te tirohanga ofi he tino faahiahia, rite homai e te ōwehenga i waenganui i te kuware me tau tōpu. I muri i te katoa, i roto i tona taha maui e whakanuia ore maha kīanga e whakawhirinaki anake i runga i ohie, a i roto i te nui e tika ana e pā ana i te tau tōpu pai katoa.

haere Rēimāna i Hēura. I roto i te tikanga ki te kitea te kī ki te raruraru o te tohatoha o te tau, Te anihia nei te reira ki te tautuhi i te tātai hoki e rua te tāupe tūturu me matatini. Ko te ia e muri mai ka mohiotia rite te Zeta mahi Rēimāna. I roto i te tau 1859 whakaputaina te kaipūtaiao, he tuhinga e tika ana "I te maha o primes kore e e nui ake i te uara tomu'a", i whakarapopototia ki o ratou whakaaro katoa.

whakaarohia Rēimāna te whakamahi o te maha o Hēura, convergent mo tūturu s katoa> 1. Ki te te whakamahia te tātai taua mo s matatini, na ka kōrero te raupapa mō tetahi uara o te tāupe ki te wahi tūturu he nui atu i 1. whakamahia Rēimāna te hokohoko atu analytic o te tukanga i whakawhänui i te whakamāramatanga o Zeta (s) mo tau matatini katoa, engari "i whiua e" kōwae. kihai hoki i ahei, no te mea ki te = s 1 piki Zeta mahi ki mutu-.

tikanga mahi

whakatika te pātai: he aha te mea ngā, me te nui mahi Zeta, i te mea faufaa i roto i te mahi o Rēimāna i runga i te whakapae korenga? Ka rite ki matau koutou, i te kau kihai i kitea te tauira ohie e whakaahua ana i te tohatoha o ngā tau pirimia i roto i te taiao. Rēimāna taea ki te kite e te maha o te pi (x) o tau pirimia, e kore e pai ki te x nei, kua whakahuatia e te tohatoha o nontrivial mahi kore Zeta. Ano, ko te whakapae Rēimāna ko te huru tika i roto i te tikanga ki te whakamatau aromātai rangitahi o etahi hātepe Tuhimunatanga.

Ko te whakapae Rēimāna

Ko tētahi o nga formulations tuatahi o tenei raruraru pāngarau, e kore kï ki tenei ra, ko: mama mahi 0 Zeta - tau matatini ki wahi tūturu rite ki te ½. I roto i te mau parau te tahi atu, e whakaritea e ratou i runga i te raina tika Re s = ½.

He he ano hoki te Rēimāna whakapae te whakatakiruatanga o, i te mea te tauākī kotahi, engari mo te whānui o te Zeta-mahi, e te hunga kua huaina te Dirichlet (kite. Photo raro) L-mahi.

I roto i te χ tātai (n) - he pūāhua tau (k mod).

tauākī a Rēimāna ko te whakapae i whakakahore na-ka karanga, Kua rite kua manatoko hoki ōritetanga ki te raraunga tauira tīariari.

Ka rite ki tohe ahau Rēimāna

i whakatakoto tuatahi Note mathematician Tiamana rawa ore. Ko te meka ko e i taua wa te pūtaiao haere i ki te whakamatau i te ture tauwehe i runga i te tohatoha o ngā tau pirimia, a i roto i tenei horopaki, e kore e tenei whakapae i te nui pānga. Heoi, tona tūranga i roto i te whakatutuki i te maha atu ngā take he nui. Ko Ko te aha te whakapae Rēimāna hoki inaianei ia ite e rave rahi aivanaa te nui o ngā raruraru pāngarau e keri.

Ko nga mea i mea, ki te whakamatau i te ture tauwehe i runga i te tohatoha o te whakapae Rēimāna tonu ko kore e tika, a tino arorau whakamatautau e te wahi tūturu o tetahi kore-mama kore o te mahi Zeta ko waenganui i 0 me 1. fakahu'unga tenei taonga e te moni o te katoa 0-m Zeta mahi e puta i roto i te tātai tangohia i runga, - tahuti tamau. Hoki uara nui o te x, e taea te ngaro katoa te reira. Ko te mema anake o te tātai, e ka noho tonu noa i x tiketike rawa, he ia x. Ko te toenga o nga ngā matatini i roto i te rite ki reira Pātapa ngaro. Ko te kupu, e whangai ana i te moni tautaha ki x. Ka taea te whakaaro i tēnei meka rite tohu o te pono o te ture tauwehe tau pirimia. Ko te kupu, ko te ngā kore o te mahi Rēimāna Zeta puta he tūranga motuhake. Ko reira ki te whakamatau e kore e taea e enei uara whai wāhi tino ki te tātai roha.

kau muimui Rēimāna

Ko te mate fakamamahi i kohi tatatia kawea te pūtaiao ki te pito arorau o te hōtaka. Heoi, ka mau e ia te tukapaku, i te W-F. de la Vallée Poussin me Zhak Adamar. Takitahi o ia atu i mawehe ratou ture tauwehe tau pirimia. whakahaere Hadamard ko Poussin ki te whakamatau i kei nontrivial katoa mahi 0 Zeta e i roto i te ropu tino.

Ngā mihi ki te mahi o enei pūtaiao, he peka hou o te pāngarau - ariā tātari o tau. I muri mai, kua riro i te tohu atu tahito iti o i mahi te ture tauwehe i roto i Roma atu kairangahau. I roto i ngā, kua whakatuwheratia Pal Erdös me Atle Selberg noa whakau tona mekameka tino matatini o arorau, e kore e rapu i te whakamahi o te tātari matatini. Heoi, i tenei wāhi te whakaaro o Rēimāna e rave rahi theorems nui kua kï, tae atu i te āwhiwhitanga o nga mahi maha o tau ariā. I roto i te hononga ki tenei mahi hou Erdős me Atle Selberg tata tetahi mea e kore e pāngia.

Ko tētahi o te taunakitanga māmā me tino ataahua o te raruraru i kitea i roto i 1980 e Donald Newman. I hāngai te reira i runga i te ture tauwehe Cauchy pai-mohiotia.

Whakawehi ki te he whakapae o Rēimāna te pūtake o te tuhiwhakamuna hou

puta whakamunatanga Raraunga ki te ahua o pūāhua, kaua, kia i titiro ai ratou ia ratou rite te waehere tuatahi ranei. I te kau, i reira te mea he au hou katoa o te tuhiwhakamuna mamati, mahi nei i roto i te whanaketanga o hātepe whakamunatanga.

Simple, me "Semisimple" tau m. E. hunga e e anake wehea ki atu e rua tau o te piha haapiiraa i taua, ko te pūtake o te pūnaha matua tūmatanui, e mohiotia ana ko RSA. He he tono whānui. I roto i ngā, kua whakamahia te reira i roto i te whakatupuranga o te waitohu hiko. Ki te kōrero tatou i roto i ngā o te wātea "teapot", i whakapuakina ai te whakapae Rēimāna te oraraa o te pūnaha i te tohatoha o ngā tau pirimia. Ko te kupu, tino heke ātete o taviri Tuhimunatanga, i runga i nei, tei te haumaru o whakawhitinga ipurangi i roto i te e-tauhokohoko.

Ētahi atu raruraru pāngarau kāore

Whakaoti tuhinga he utu rawatia a he torutoru kupu ki ētahi atu mahi o te mano tau. Ko ētahi o ēnei:

• Takatāpui o ngā akomanga P me NP. whakatakoto te raruraru te e whai ake: Ki te manatoko i te whakahoki pai ki te pātai i homai i roto i te wa pūrau, na ko reira pono e ia ia te whakahoki ki tēnei pātai e taea e kitea hohoro?
• Hodge tāpae kōrero. I roto i ngā ohie e taea te kī i te reira e whai ake: hoki etahi momo o manifolds taurangi projective (mokowā) Hodge huringa he ngā o ngā e whai he tikanga āhuahanga, arā huringa taurangi ...
• tāpae kōrero Poincaré. Ko kï i nga raruraru kau mano tau te anake. E ai ki reira tetahi ahanoa e toru-ahu he āhuatanga motuhake o te ao-ahu 3, me kia tika ki te deformation te ao.
• Whakaaetanga o te quantum Yang - Mills ariā. E ti'a ia tatou ki te whakamatau i taua aronui ariā, hoatu mua i enei pūtaiao ki te R wāhi ^4, i reira te mea he koha tona 0-papatipu mō tetahi whakatika ohie o te rōpū whāiti G.
• Ko te whakapae o te Birch - Swinnerton-Dyer. Ko tetahi raruraru i te mea e hāngai ana ki te Tuhimunatanga tenei. āwangawanga te reira i te ānau porotītaha.
• Ko te raruraru o te oraraa, me te whakapati o rongoā o te Navier - whārite Stokes.

Na e matau ana koutou ki te whakapae Rēimāna. I roto i ngā ohie, kua whakatakoto matou me etahi o te tahi atu whāinga o te mano tau. Ko te take i ka whakatau ratou ranei e whakamatauria reira e whai ratou kahore otinga - te reira i te mea o te wā. A he pea ki te whai ki te tatari roa rawa, rite e piki nga pāngarau te whakamahi i te mana tātai o rorohiko tenei. Heoi, he kaupapa ki te toi e kore nga mea katoa, me te ki te whakaoti rapanga pūtaiao matua titau pūmanawa me te auaha.