Lub Riemann hypothesis. Faib prime xov tooj

Nyob rau hauv 1900, ib tug ntawm cov loj tshaj zaum ntawm lub xyoo pua xeem, David Hilbert ua ib daim ntawv teev hom 23 unsolved teeb meem ntawm kev kawm txog zauv. Ua hauj lwm rau lawv tau muaj ib tug zoo kawg li tej yam nyob rau hauv txoj kev loj hlob ntawm daim teb no ntawm tib neeg kev txawj ntse. Tom qab 100 xyoo nyob rau hauv lub av nplaum lej koom haum hais ib daim ntawv teev cov xya teeb meem, lub npe hu ua lub xyoo txhiab hom phiaj. Rau qhov kev txiav txim ntawm txhua tus ntawm lawv twb muaj cov nqi zog ntawm $ 1 million.

Cov teeb meem nkaus xwb, uas yog cov ob lub npe ntawm learn, rau centuries tsis tau muab so rau zaum, los ua tus Riemann hypothesis. Nws yog tseem tos rau nws txiav txim siab.

Luv luv biographical ntaub ntawv

Georg Friedrich Bernhard Riemann yug nyob rau hauv 1826 nyob rau hauv Hanover, nyob rau hauv ib tug loj tsev neeg ntawm ib tug neeg txom nyem ib tug xibhwb, thiab ua neej nyob tau 39 xyoo. Nws muaj kev tswj los mus luam 10 cov ntaub ntawv. Txawm li cas los, thaum lub sij hawm lub neej ntawm Riemann nws suav hais tias ib tug successor ntawm nws tus kws qhia ntawv Johann Gauss. Thaum 25 xyoo cov tub ntxhais paub txog defended nws thesis "me nyuam yaus ntawm txoj kev tshawb xav ntawm kev khiav dej num ntawm ib tug complex nce mus nce los." Tom qab ntawd nws formulated nws hypothesis, uas ua nto moo.

primes

Mathematics tuaj thaum tus txiv neej kawm suav. Ces sawv thawj lub tswv yim ntawm cov zauv, uas tom qab sim cais. Nws tau raug pom tias ib txhia ntawm lawv muaj ib qho zog. Nyob rau hauv kev, cov qhov ntuj tooj m. E. Cov uas tau siv nyob rau hauv lub xam (numbering) los yog cov hoob nab npawb ntawm cov khoom tau tau faib ib pab pawg neeg ntawm xws uas yog muab faib los ntawm ib tug thiab lawv tus kheej. Lawv hu ua yooj yim. Ib tug elegant pov thawj ntawm cov theorem infinite set ntawm cov zauv uas muab los ntawm Euclid nyob rau hauv nws "hais". Thaum lub caij, peb yuav ntxiv lawv nrhiav. Nyob rau hauv kev, qhov loj tshaj plaws ntawm ib tug xov tooj ntawm cov paub ² 74207281 - 1.

Euler tus mis

Nrog rau lub notion ntawm infinitely ntau primes Euclid txhais thiab lub thib ob theorem tsuas tau factorization. Raws li nws tej zoo integer yog qhov khoom ntawm tsuas yog ib co ntawm cov primes. Nyob rau hauv 1737, tus poj German mathematician Leonhard Euler qhia thawj ntawm Euclid lub theorem rau lub infinity ntawm lub mis qhia hauv qab no.

Nws yog hu ua tus zeta muaj nuj nqi, qhov twg s - ib qhov thiab p yog tag nrho tej yam yooj yim qhov tseem ceeb. Los ntawm nws ncaj qha ua raws li thiab pom zoo los ntawm lub uniqueness ntawm lub expansion ntawm Euclid.

Riemann zeta muaj nuj nqi

Euler tus formula rau closer soj ntsuam yog heev zoo kawg li, raws li muab los ntawm tus piv ntawm cov yooj yooj yim thiab zauv. Tom qab tag nrho cov, nyob rau hauv nws sab laug yog multiplied infinitely ntau kab zauv uas yog nyob tsuas yog nyob rau tej yam yooj yim, thiab nyob rau hauv txoj cai npaum li cas yog txuam nrog tag nrho cov zoo cov zauv.

Riemann mus rau Euler. Nyob rau hauv thiaj li yuav nrhiav tus yuam sij rau qhov teeb meem ntawm lub tsev ntawm cov zauv, nws yog npaj los txhais tau lub mis rau ob lub tiag tiag thiab complex nce mus nce los. Nws yog nws leej twg tom qab lub npe hu ua lub Riemann zeta muaj nuj nqi. Nyob rau hauv 1859 tus paub txog luam tawm ib tsab xov xwm hu ua "Nyob rau hauv lub xov tooj ntawm primes uas tsis pub tshaj ib tug predetermined nqi", uas summed li tag nrho lawv tej tswv yim.

Riemann npaj rau kev siv ntawm ib tug xov tooj ntawm Euler, convergent rau tag nrho cov tiag tiag s> 1. Yog hais tias tib lub mis yog siv rau txoj s, ces tus series yuav sib rau tej nqi ntawm qhov kev nce mus nce los nrog lub tiag tiag ib feem yog ntau tshaj 1. Riemann siv cov analytic continuation ntawm cov txheej txheem los ntawm expanding lub ntsiab txhais ntawm zeta (s) rau tag nrho cov lej, tab sis "ntuav" unit. Nws twb tsis tau, vim hais tias yog s = 1 zeta muaj nuj nqi nce mus infinity.

tswv yim kev txiav txim zoo

Cov lus nug tshwm sim: dab tsi yog nthuav thiab tseem ceeb zeta muaj nuj nqi, uas yog tseem ceeb heev nyob rau hauv lub chaw ua hauj lwm ntawm Riemann nyob rau null hypothesis? Raws li koj paub, thaum lub caij tsis pom ib tug yooj yim qauv uas qhia txog lub tsev ntawm prime xov tooj ntawm cov natural. Riemann tau los xyuas hais tias cov xov tooj ntawm pi (x) ntawm prime xov tooj, uas yog tsis superior rau x, yog qhia los ntawm lub tsev ntawm nontrivial pes tsawg zeta muaj nuj nqi. Ntxiv mus, lub Riemann hypothesis yog ib tug tsim nyog mob nyob rau hauv thiaj li yuav ua pov thawj ib ntus ntsuam xyuas ntawm tej yam cryptographic algorithms.

Lub Riemann hypothesis

Ib qho ntawm thawj formulations ntawm no xyuam xim teeb meem, tsis muaj pov thawj rau hnub no, yog: tsis tseem ceeb 0 zeta muaj nuj nqi - lej nrog tiag tiag ib feem sib npaug zos rau ½. Nyob rau hauv lwm yam lus, lawv cov txheej txheem ntawm nyob rau hauv ib tug ncaj kab Re s = ½.

Muaj kuj yog ib tug generalized Riemann hypothesis, uas yog tib yam daim ntawv, tab sis rau generalization ntawm lub zeta-zog, uas yog hu ua lub Dirichlet (saib. Photo hauv qab no) L-zog.

Nyob rau hauv cov mis χ (n) - ib tug zauv cim (mod k).

Riemann cov lus yog li no-hu ua thov hypothesis, raws li tau qhia tau tseeb rau taub hau nrog tus uas twb muaj qauv ntaub ntawv.

Raws li kuv sib cav Riemann

Ceeb toom German mathematician yog Ameslikas formulated heev casually. Qhov tseeb yog tias nyob rau hauv hais tias lub sij hawm tus paub txog yuav los ua pov thawj ib tug theorem rau lub tsev ntawm prime numbers, thiab nyob rau hauv qhov ntsiab lus teb no, qhov no hypothesis tsis muaj ntau cov nyhuv. Txawm li cas los, nws lub luag hauj lwm nyob rau hauv qhov ntau lwm yam teeb meem yog enormous. Uas yog vim li cas lub Riemann hypothesis rau tam sim no muaj ntau zaum paub txog qhov tseem ceeb ntawm meej zauv teeb meem.

Raws li tau hais tias, ua pov thawj rau theorem rau lub tsev ntawm lub tag nrho Riemann hypothesis yog tsis tsim nyog, thiab heev txawj ua pov thawj hais tias tus tiag tiag yog ib feem ntawm tej uas tsis yog-tsis tseem ceeb pes tsawg ntawm cov zeta muaj nuj nqi yog nruab nrab ntawm 0 thiab 1. Qhov no tej khoom vaj tse implies hais tias cov sum ntawm tag nrho cov 0-m zeta muaj nuj nqi uas tshwm sim nyob rau hauv lub caij nyoog mis saum toj no, - finite qhov. Rau loj qhov tseem ceeb ntawm x, nws muaj peev xwm tag nrho yuav raug poob lawm. Cov tib tug mej zeej ntawm lub mis, uas yuav nyob twj ywm unchanged txawm nyob rau hauv heev x, x yog nws tus kheej. Tus so ntawm lub complex cov ntsiab lus nyob rau hauv kev sib piv nrog nws asymptotically ploj. Yog li, lub weighted sum nyhav x. Qhov tseeb yuav suav hais tias raws li pov thawj ntawm qhov tseeb ntawm prime tooj theorem. Yog li, lub zeros ntawm lub Riemann zeta muaj nuj nqi zoo nkaus li ib tug tshwj xeeb lub luag hauj lwm. Nws yog los ua pov thawj hais tias cov qhov tseem ceeb yuav tsis ua ho mus rau lub expansion mis.

Riemann followers

Lub tragic kev tuag los ntawm tus kab mob ntsws tiv thaiv tau tus paub txog coj mus rau cov zajlus kom xaus ntawm qhov kev pab cuam. Txawm li cas los, nws coj lub baton los ntawm cov W-F. de la Vallée Poussin thiab Zhak Adamar. Ntawm nws tus kheej ntawm txhua lwm yam lawv tau rho prime tooj theorem. Hadamard thiab Poussin tswj los ua pov thawj hais tias tag nrho cov nontrivial 0 zeta muaj nuj nqi yog nyob hauv lub tseem ceeb heev band.

Tsaug rau cov hauj lwm ntawm cov zaum, ib tug tshiab ceg ntawm kev kawm txog zauv - analytical kev tshawb xav ntawm cov zauv. Tom qab ntawd, lwm yam kev soj ntsuam ntawm tau txais ib tug me ntsis ntau cov tswv yim qub pov thawj ntawm cov theorem tau ua hauj lwm nyob rau hauv lub nroog Loos. Nyob rau hauv kev, npawg Erdös thiab Atle Selberg tau qhib txawm lees tias nws mas complex saw ntawm logic, tsis yuav tsum tau siv ntawm complex tsom xam. Txawm li cas los, nyob rau ntawm no point lub tswv yim ntawm Riemann los ntawm ob peb yam tseem ceeb theorems tau pov thawj, nrog rau cov approximation ntawm ntau zog ntawm kev tshawb xav tooj. Nyob rau hauv kev twb kev txuas nrog rau qhov no tshiab ua hauj lwm Erdős thiab Atle Selberg zoo dab tsi tsis cuam tshuam.

Ib tug ntawm cov nyuaj thiab zoo nkauj tshaj plaws pov thawj ntawm qhov teeb meem tau raug muaj nyob rau hauv 1980 los ntawm Donald Newman. Nws twb raws li nyob rau hauv lub zoo-paub Cauchy theorem.

Hawv tias Riemann tus hypothesis yog lub hauv paus ntawm niaj hnub cryptography

Cov ntaub ntawv encryption sawv nrog lub tsos ntawm cov cim, los yog es, lawv lawv tus kheej yuav tsum tau ntshai raws li cov thawj code. Thaum lub caij, muaj yog ib tug tseem tshiab sib ntawm cov cryptography, uas yog koom nyob rau hauv txoj kev loj hlob ntawm encryption algorithms.

Tej yam yooj yim thiab "Semisimple" tooj m. E. Cov uas yog tsuas muab faib ua ob tug lwm cov xov tooj ntawm cov tib chav kawm ntawv, yog lub hauv paus ntawm ib tug pej xeem tseem ceeb system, lub npe hu ua RSA. Nws muaj ib tug ntau daim ntawv thov. Nyob rau hauv kev, nws yog siv nyob rau hauv lub cim ntawm ib tug hluav taws xob kos npe. Yog hais tias peb tham nyob rau hauv cov nqe lus ntawm cov muaj "lauj kaub rau ces", lub Riemann hypothesis siv lub hav zoov ntawm cov system nyob rau hauv lub tsev ntawm prime xov tooj. Yog li, ho txo kuj cryptographic tuav, uas nyob ruaj ntseg ntawm online muas nyob rau hauv e-lag luam.

Lwm yam unsolved zauv teeb meem

Sau kom tiav tsab xov xwm no muaj nqis siab ntso ib tug ob peb lo lus rau lwm yam ntawm lub xyoo txhiab. Cov muaj xws li:

• Koob pheej ntawm lawv cov chav kawm ntawv P thiab NP. Qhov teeb meem yog formulated raws li nram no: yog hais tias ib tug zoo lus teb rau ib tug muab lus nug no yog muaj tseeb nyob rau hauv polynomial lub sij hawm, ces yog nws muaj tseeb hais tias nws tus kheej cov lus teb rau lo lus nug no yuav tsum pom sai sai?
• Hodge conjecture. Nyob rau hauv tej yam yooj yim cov ntsiab lus uas nws yuav tau teev raws li nram no: rau ib co hom projective algebraic manifolds (tej qhov chaw) Hodge mus yog ob peb ua ke ntawm cov khoom uas muaj ib tug duab txhais, piv txwv li algebraic mus ...
• Poincaré conjecture. Nws yog ib lub xwb qhov tseeb thaum lub caij xyoo txhiab teeb meem. Raws li nws tej peb-dimensional khoom muaj tej thaj chaw ntawm lub 3-dimensional kheej, tus kheej yuav tsum yog muaj tseeb rau deformation.
• Pom zoo los ntawm lub quantum Yang - Mills kev tshawb xav. Peb yuav tsum tau los ua pov thawj hais tias quantum kev tshawb xav, muab tso rau pem hauv ntej los ntawm cov zaum mus rau lub chaw R ^4, yog ib tug 0-loj defect rau tej yam yooj yim calibration ntawm ib tug compact pab pawg neeg G.
• Cov hypothesis ntawm lub Birch - Swinnerton-Dyer. Qhov no yog lwm qhov teeb meem uas yog cov mus cryptography. Nws kev txhawj xeeb hauv lub elliptical curves.
• Qhov teeb meem ntawm lub hav zoov thiab smoothness ntawm kev daws teeb meem ntawm lub Navier - Stokes sib npaug.

Tam sim no koj paub lub Riemann hypothesis. Nyob rau hauv tej yam yooj yim, peb muaj formulated thiab ib co ntawm lwm yam hom phiaj ntawm lub xyoo txhiab. Qhov tseeb hais tias lawv yuav tsum tau muab daws los yog nws yog muaj pov thawj hais tias lawv muaj tsis muaj tshuaj - nws yog ib tug teeb meem ntawm lub sij hawm. Thiab qhov no yog tsis zoo li yuav tau tos ntev ntev, raws li lub lej yog nce siv computational hwj chim ntawm computers. Txawm li cas los, tsis yog txhua yam kev kawm rau cov kos duab thiab los daws kom tau scientific teeb meem feem ntau yuav tsum tau qhov thiab creativity.