Nyob rau xyoo 1900, yog ib tus kws tshawb fawb loj tshaj plaws ntawm lub xyoo pua xeem, David Hilbert, tau sau cov npe ntawm 23 qhov teeb meem tsis tau daws hauv kev ua lej. Kev ua haujlwm ntawm lawv tau muaj kev cuam tshuam loj heev rau kev txhim kho ntawm thaj chaw no ntawm tib neeg kev paub. 100 xyoo tom qab ntawd, lub koom haum Clay Mathematical Institute tau nthuav tawm cov npe ntawm 7 qhov teeb meem hu ua Millennium Problems. Txhua tus tau muab nqi zog $1 lab.
Qhov teeb meem nkaus xwb uas tau tshwm sim ntawm ob daim ntawv teev cov kev sib tw uas tau haunting cov kws tshawb fawb rau ntau tshaj ib puas xyoo yog Riemann kev xav. Nws tseem tos nws qhov kev txiav txim.
Cov ntawv luv luv
Georg Friedrich Bernhard Riemann yug hauv 1826 hauv Hannover, hauv ib tsev neeg loj ntawm ib tug xibhwb pluag, thiab nyob tsuas yog 39 xyoo. Nws tswj tau tshaj tawm 10 tej hauj lwm. Txawm li cas los xij, twb nyob rau hauv nws lub neej, Riemann tau suav hais tias yog tus thawj coj ntawm nws tus kws qhia Johann Gauss. Thaum muaj hnub nyoog 25 xyoos, tus kws tshawb fawb hluas tau tiv thaiv nws cov lus tshaj tawm "Lub hauv paus ntawm txoj kev xav ntawm kev ua haujlwm ntawm kev sib txawv." Tom qab ntawd nws formulatednws nto moo hypothesis.
lej tshwm sim thaum txiv neej kawm suav. Nyob rau tib lub sijhawm, thawj cov tswv yim hais txog cov lej tau tshwm sim, uas tom qab ntawd lawv tau sim cais tawm. Qee tus ntawm lawv tau pom tias muaj cov khoom sib xws. Tshwj xeeb, ntawm cov lej ntuj, piv txwv li, cov uas tau siv los suav (ntaus) lossis xaiv tus naj npawb ntawm cov khoom, ib pab pawg tau sib txawv uas tau faib los ntawm ib leeg thiab ntawm lawv tus kheej. Lawv hu ua yooj yim. Ib qho pov thawj zoo nkauj ntawm lub theorem ntawm infinity ntawm cov txheej txheem ntawm cov lej tau muab los ntawm Euclid hauv nws Cov Ntsiab Lus. Lub sijhawm no, lawv txoj kev tshawb nrhiav txuas ntxiv mus. Tshwj xeeb, tus lej loj tshaj plaws uas twb paub lawm yog 274 207 281 - 1.
Euler formula
Nrog rau lub tswv yim ntawm infinity ntawm cov txheej txheem ntawm primes, Euclid kuj tau txiav txim siab qhov thib ob theorem ntawm qhov tsuas muaj peev xwm decomposition mus rau hauv thawj yam tseem ceeb. Raws li nws, txhua tus lej zoo yog cov khoom lag luam ntawm tsuas yog ib pawg ntawm tus lej tseem ceeb. Xyoo 1737, tus kws lej German zoo Leonhard Euler tau hais txog Euclid thawj qhov kev xav tsis kawg raws li cov qauv hauv qab no.
Nws yog hu ua zeta muaj nuj nqi, qhov twg s yog qhov tas li thiab p siv tag nrho cov txiaj ntsig tseem ceeb. Euclid cov lus hais txog qhov tshwj xeeb ntawm kev nthuav dav ncaj qha los ntawm nws.
Riemann Zeta Function
Euler tus qauv, ntawm kev soj ntsuam ze dua, yog kiag lixav tsis thoob vim nws txhais cov kev sib raug zoo ntawm primes thiab integers. Tom qab tag nrho, infinitely ntau kab lus uas nyob ntawm seb tsuas yog cov lej tseem ceeb tau muab faib rau ntawm nws sab laug, thiab cov lej cuam tshuam nrog txhua tus lej zoo nyob rau sab xis.
Riemann mus ntxiv dua Euler. Txhawm rau txhawm rau nrhiav tus yuam sij rau qhov teeb meem ntawm kev faib cov lej, nws tau thov kom txhais cov qauv rau ob qho tib si tiag tiag thiab qhov sib txawv. Nws yog nws uas tom qab tau txais lub npe ntawm Riemann zeta muaj nuj nqi. Xyoo 1859, tus kws tshawb fawb tau luam tawm ib tsab xov xwm hu ua "Ntawm tus lej ntawm tus lej tseem ceeb uas tsis pub tshaj tus nqi", uas nws tau sau tag nrho nws cov tswv yim.
Riemann tau hais kom siv Euler series, uas sib koom ua ke rau ib qho tiag s>1. Yog hais tias tib lub mis yog siv rau complex s, ces cov series yuav converge rau tej tus nqi ntawm no variable nrog ib tug tiag tiag ib feem ntau dua 1. Riemann siv cov txheej txheem analytic continuation, ncua lub ntsiab lus ntawm zeta(s) rau tag nrho cov complex zauv, tab sis. "tso tawm" unit. Nws raug cais tawm vim hais tias ntawm s=1 lub zeta muaj nuj nqi nce mus rau infinity.
Kev nkag siab zoo
Ib lo lus nug tshwm sim: vim li cas lub zeta muaj nuj nqi, uas yog qhov tseem ceeb hauv Riemann txoj haujlwm ntawm qhov kev xav tsis zoo, nthuav thiab tseem ceeb? Raws li koj paub, tam sim no tsis muaj tus qauv yooj yim tau txheeb xyuas uas yuav piav qhia txog kev faib cov lej tseem ceeb ntawm cov lej ntuj. Riemann tau tshawb pom tias tus lej pi(x) ntawm primes uas tsis tshaj x yog qhia nyob rau hauv cov nqe lus ntawm kev faib cov tsis tseem ceeb zeros ntawm lub zeta muaj nuj nqi. Ntxiv mus, Riemann hypothesis yogib qho kev tsim nyog rau kev ua pov thawj lub sij hawm kwv yees rau kev ua haujlwm ntawm qee qhov cryptographic algorithms.
Riemann Hypothesis
Ib qho ntawm thawj cov qauv ntawm qhov teeb meem kev ua lej, uas tsis tau muaj pov thawj rau hnub no, suab zoo li no: cov haujlwm tsis tseem ceeb 0 zeta yog cov lej nyuaj nrog qhov sib npaug ntawm ½. Hauv lwm lo lus, lawv nyob ntawm kab Re s=½.
Tseem kuj tseem muaj qhov kev xav dav dav ntawm Riemann, uas yog tib nqe lus, tab sis rau kev nthuav dav ntawm zeta zog, uas feem ntau hu ua Dirichlet L-functions (saib daim duab hauv qab no).
Nyob hauv cov mis χ(n) - qee tus lej cim (modulo k).
Cov lus Riemannian yog suav tias yog lub npe hu ua null hypothesis, vim nws tau raug sim rau qhov sib xws nrog cov ntaub ntawv uas twb muaj lawm.
Raws Riemann tau sib cav
Cov lus hais ntawm tus kws lej German yog thawj lo lus zoo li casually. Qhov tseeb yog tias lub sijhawm ntawd tus kws tshawb fawb tau mus ua pov thawj lub theorem ntawm kev faib cov lej tseem ceeb, thiab hauv cov ntsiab lus no, qhov kev xav no tsis muaj qhov tseem ceeb. Txawm li cas los xij, nws lub luag haujlwm hauv kev daws teeb meem ntau ntxiv yog qhov loj heev. Tias yog vim li cas Riemann qhov kev xav tam sim no tau lees paub los ntawm ntau tus kws tshawb fawb tias yog qhov tseem ceeb tshaj plaws ntawm cov teeb meem lej uas tsis muaj pov thawj.
Raws li twb tau hais lawm, tag nrho Riemann lub tswv yim tsis xav tau los ua pov thawj qhov kev faib tawm, thiab nws txaus los ua pov thawj tias qhov tseeb ntawm ib qho tsis tseem ceeb ntawm xoom ntawm zeta muaj nuj nqi yog nyob rau hauvnruab nrab ntawm 0 thiab 1. Nws ua raws los ntawm cov cuab yeej no tias cov sum tshaj tag nrho 0's ntawm lub zeta muaj nuj nqi uas tshwm nyob rau hauv lub pes tsawg tus qauv saum toj no yog ib tug finite tas li. Rau qhov loj ntawm x, nws yuav ploj tag nrho. Tsuas yog cov tswv cuab ntawm cov mis uas tseem zoo ib yam txawm tias loj heev x yog x nws tus kheej. Cov ntsiab lus nyuaj ntxiv ploj mus asymptotically piv nrog nws. Yog li ntawd, qhov sib npaug ntawm qhov hnyav nyhav rau x. Qhov xwm txheej no tuaj yeem suav tias yog kev lees paub qhov tseeb ntawm qhov theorem ntawm kev faib cov lej tseem ceeb. Yog li, lub zeros ntawm Riemann zeta muaj nuj nqi muaj lub luag haujlwm tshwj xeeb. Nws muaj nyob rau hauv kev ua pov thawj tias cov txiaj ntsig zoo li no tsis tuaj yeem ua rau muaj txiaj ntsig zoo rau cov qauv decomposition.
Cov neeg ntawm Riemann
Kev tuag los ntawm tuberculosis tsis tau tso cai rau tus kws tshawb fawb no coj nws txoj haujlwm rau nws qhov laj thawj kawg. Txawm li cas los xij, Sh-Zh tau los ntawm nws. de la Vallée Poussin thiab Jacques Hadamard. Independent ntawm ib leeg, lawv deduced ib tug theorem ntawm kev faib cov zauv. Hadamard thiab Poussin tau tswj kom ua pov thawj tias txhua yam tsis tseem ceeb 0 zeta ua haujlwm nyob hauv pawg tseem ceeb.
Ua tsaug rau cov haujlwm ntawm cov kws tshawb fawb no, cov kev taw qhia tshiab hauv kev ua lej tau tshwm sim - txoj kev xav ntawm cov lej. Tom qab ntawd, ntau qhov pov thawj tseem ceeb ntawm lub theorem uas Riemann tau ua haujlwm tau txais los ntawm lwm cov kws tshawb fawb. Hauv particular, Pal Erdős thiab Atle Selberg txawm nrhiav tau ib txoj hauv kev yooj yim heev uas lees paub nws, uas tsis tas yuav tsum tau siv cov kev tshuaj xyuas nyuaj. Txawm li cas los xij, los ntawm qhov no, ntau qhov tseem ceebtheorems, suav nrog kev kwv yees ntawm ntau txoj kev xav ntawm tus lej. Hauv qhov no, txoj haujlwm tshiab ntawm Erdős thiab Atle Selberg xyaum tsis cuam tshuam dab tsi.
Ib qho yooj yim thiab zoo nkauj tshaj pov thawj ntawm qhov teeb meem tau pom nyob rau xyoo 1980 los ntawm Donald Newman. Nws yog raws li lub npe nrov Cauchy theorem.
Puas Riemannian kev xav txog kev hem thawj ntawm lub hauv paus ntawm niaj hnub cryptography
Cov ntaub ntawv encryption tshwm sim nrog rau cov tsos ntawm hieroglyphs, ntau dua, lawv tus kheej tuaj yeem suav tias yog thawj cov lej. Tam sim no, muaj tag nrho cheeb tsam ntawm cov lej cryptography, uas tab tom tsim encryption algorithms.
Prime thiab "semi-prime" tus lej, piv txwv li cov uas tsuas yog faib tau los ntawm 2 lwm tus lej los ntawm tib chav kawm, tsim lub hauv paus ntawm pej xeem cov txheej txheem tseem ceeb hu ua RSA. Nws muaj daim ntawv thov dav tshaj plaws. Tshwj xeeb, nws yog siv thaum tsim hluav taws xob kos npe. Hais lus nyob rau hauv cov nqe lus siv tau rau dummies, lub Riemann hypothesis hais tias muaj ib tug system nyob rau hauv kev faib ntawm cov xov tooj tseem ceeb. Yog li, lub zog ntawm cov yuam sij cryptographic, uas qhov kev ruaj ntseg ntawm kev lag luam online hauv kev lag luam e-lag luam nyob ntawm, tau txo qis.
Lwm yam tsis daws teeb meem lej
Nws tsim nyog ua kom tiav tsab xov xwm los ntawm kev mob siab rau ob peb lo lus rau lwm lub hom phiaj xyoo txhiab. Cov no suav nrog:
- Kev sib npaug ntawm chav kawm P thiab NP. Qhov teeb meem yog tsim raws li hauv qab no: yog tias cov lus teb zoo rau cov lus nug tshwj xeeb raug kuaj xyuas hauv lub sijhawm polynomial, ces nws puas muaj tseeb tias cov lus teb rau lo lus nug no nws tus kheejnrhiav tau sai?
- Hodge qhov kev xav. Hauv cov lus yooj yooj yim, nws tuaj yeem tsim tau raws li hauv qab no: rau qee hom phiaj xwm algebraic ntau yam (chaw), Hodge cycles yog kev sib txuas ntawm cov khoom uas muaj kev txhais geometric, piv txwv li, algebraic cycles.
- Poincaré qhov kev xav. Qhov no tsuas yog Millennium Challenge uas tau ua pov thawj txog tam sim no. Raws li nws, txhua yam khoom 3-dimensional uas muaj cov khoom tshwj xeeb ntawm 3-dimensional sphere yuav tsum yog tus kheej kheej, mus txog deformation.
- Kev lees paub ntawm quantum theory of Yang - Mills. Nws yuav tsum tau ua pov thawj tias quantum txoj kev xav tau muab tso rau los ntawm cov kws tshawb fawb no rau qhov chaw R 4 muaj thiab muaj qhov tsis zoo 0th rau ib qho yooj yim compact gauge pawg G.
- Birch-Swinnerton-Dyer hypothesis. Qhov no yog lwm qhov teeb meem ntsig txog cryptoography. Nws kov elliptic nkhaus.
- Qhov teeb meem ntawm qhov muaj nyob thiab smoothness ntawm kev daws rau Navier-Stokes equations.
Tam sim no koj paub Riemann kev xav. Hauv cov ntsiab lus yooj yim, peb tau tsim qee qhov ntawm lwm qhov Kev Sib Tw Millennium. Tias lawv yuav daws tau lossis nws yuav raug pov thawj tias lawv tsis muaj kev daws teeb meem yog lub sijhawm. Tsis tas li ntawd, nws tsis zoo li qhov no yuav tsum tau tos ntev heev, txij li kev ua lej yog nce siv lub peev xwm ntawm cov khoos phis tawj. Txawm li cas los xij, tsis yog txhua yam yuav raug siv thev naus laus zis, thiab ua ntej ntawm tag nrho, qhov kev xav thiab kev muaj tswv yim yuav tsum tau daws teeb meem kev tshawb fawb.
