Изворни код странице Риманова хипотеза

[[Датотека:RiemannCriticalLine.svg|thumb|Реални део (у црвеној) и имагинарни део (у плавој боји) Риманове зета-функције на критичној линији Re(''s'') = 1/2. Првих неколико имагинарних нула се може видети за вредности Im(''s'') = ±14,135, ±21,022 и ±25,011.]]

'''Риманова хипотеза''' је претпоставка о дистрибуцији нетривијалних нула [[Риманова зета-функција|Риманове зета-функције]] <math>\zeta(s)\,</math>. Први пут је формулисана у раду [[Бернард Риман|Бернарда Римана]] из 1859: ''О броју простих бројева испод задате величине'' ({{јез-нем|Über der Anzahl der Primzahlen unter einer gegebenen Größe}}). Од тада, и поред огромних напора, овај проблем и даље остаје нерешен.

Риманова зета-функција је дефинисана за све [[Комплексан број|комплексне бројеве]] ''-{s}-'' ≠ 1, и има тривијалне нуле у парним негативним целим бројевима (-{''s'' = &minus;2, ''s'' = &minus;4, ''s'' = &minus;6, ...}-). Риманова хипотеза каже да се све нетривијалне нуле налазе на једној правој у комплексној равни, конкретно:

:Реални део било које нетривијалне нуле Риманове зета-функције је &frac12;, односно све нетривијалне нуле се налазе на критичној линији &frac12; + ''-{it}-''.

== Историјат ==
Рад из [[1859]]. је Риманов једини оглед у [[теорија бројева|теорији бројева]], али је хипотеза изнета у њему један од најзначајнијих нерешених проблема у савременој [[Математика|математици]], пре свега зато што се доста важних резултата ослања на важење ове хипотезе (рецимо у [[Криптографија|криптографији]], факторизацији целих бројева и полинома).

Легенда каже да се копија сакупљених Риманових радова у Хурвицовој ({{јез-енг|Adolf Hurwitz}}) библиотеци након његове смрти сама отварала на страни на којој се налазио исказ Риманове хипотезе.

[[Давид Хилберт]] је на Другом међународном конгресу математичара у Паризу, [[8. август]]а [[1900|1900.]] године поставио проблем Риманове хипотезе као један од двадесеттри [[Хилбертови проблеми|Хилбертова проблема]] (проблем број осам). За Хилберта је Риманова хипотеза имала посебан значај, када су га питали шта би најпре урадио након 500-годишњег сна, Хилберт је одговорио да би прво питао да ли је Риманова хипотеза доказана.

Годфри Харолд Харди ({{јез-енг|Godfrey Harold Hardy}}) је [[1914|1914.]] године доказао да се на критичној линији &frac12; + ''-{it}-'' налази бесконачно много нула.

Риманова хипотеза је један од седам [[Миленијумски проблеми|Миленијумских проблема]] Математичког института Клеј.

== Покушаји доказивања ==
Риманова хипотеза је као и [[Последња Фермаова теорема]] била инспирација за небројене покушаје доказивања, где су подједнако неуспешни били и врхунски и математичари аматери. Када је 1995. године енглески математичар [[Ендру Вајлс]] извео доказ Фермаове последње теореме - фокус математичке заједнице је преусмерен на Риманову хипотезу, најистакнутији нерешени проблем у математици данас. Овде су набројани значајни неуспешни покушаји у новом миленијуму.

[[Мати Питканен]] (-{Matti Pitkanen}-) у септембру 2001, повукао доказ због грешке у новембру исте године.<ref>{{cite journal |author= |date= |title=Мати Питканен (-{Matti Pitkanen}-), Корак ближе доказу Риманове хипотезе |url= |journal= |volume=|issue= |pages= |doi= |arxiv=math/0109072}}, Приступљено 24. 4. 2013.</ref>

[[Карлос Кастро]] (-{Carlos Castro}-), и Хорге Махеха (-{Jorge Mahecha}-) су у серији радова од 2001. до 2006. године пробали да изграде теорију (користећи суперсиметрије и [[Квантна механика|квантномеханички]] приступ) која би омогућила доказивање Риманове хипотезе. Њихов приступ је одбачен.<ref>{{cite journal |author= |date= |title=Карлос Кастро (-{Carlos Castro}-) и Хорге Махеха (-{Jorge Mahecha}-), Последњи кораци ка доказу Риманове хипотезе |url= |journal= |volume=|issue= |pages= |doi= |arxiv=hep-th/0208221}}, Приступљено 24. 4. 2013.</ref>

[[Каида Ши]] (-{Kaida Shi}-) у јулу 2003. године, доказ садржавао грешку.<ref>{{cite journal |author= |date= |title=Каида Ши (-{Kaida Shi}-), Геометријски доказ уопштене Риманове хипотезе |url= |journal= |volume=|issue= |pages= |doi= |arxiv=math/0308001}}, Приступљено 24. 4. 2013.</ref>

[[Луј д'Бранж]] (-{Louis de Branges de Bourcia}-) у јулу 2004. године, нађен контрапример.<ref>{{cite web|author= |url=http://www.math.purdue.edu/~branges/riemann-hecke.pdf |title=Луј д'Бранж (-{Louis de Branges de Bourcia}-), Риманова зета-функција |date= |website= |publisher= |access-date=24. 4. 2013}}</ref> Аутор је касније објавио ''Извињење за доказ Риманове Хипотезе''.<ref>{{cite web|author= |url=http://www.math.purdue.edu/~branges/apology.pdf |title=Луј д'Бранж (-{Louis de Branges de Bourcia}-), Извињење за доказ Риманове хипотезе |date= |website= |publisher= |access-date=24. 4. 2013}}</ref>

[[Јинжу Хан]] (-{Jinzhu Han}-) у јуну 2007. године, доказ садржавао грешку.<ref>{{cite journal |author= |date= |title=Јинжу Хан (-{Jinzhu Han}-), Истраживање осмог Хилбертовог проблема |url= |journal= |volume=|issue= |pages= |doi= |arxiv=0706.1929}}, Приступљено 24. 4. 2013.</ref>

[[Андреј Мадрецки]] (-{Andrzej Madrecki}-) у јулу 2007. године, доказ садржавао грешку.<ref>{{cite journal |author= |date= |title=Андреј Мадрецки (-{Andrzej Madrecki}-), Кратак доказ Риманове хипотезе преко Брауновог кретања |url= |journal= |volume=|issue= |pages= |doi= |arxiv=0707.4196}}, Приступљено 24. 4. 2013.</ref>

[[Лев Аизенберг]] (-{Lev Aizenberg}-) у децембру 2007. године, повукао доказ због грешке у јануару 2008. године.<ref>{{cite journal |author= |date= |title=Лев Аизенберг (-{Lev Aizenberg}-), Линделофова хипотеза је тачна док Риманова није |url= |journal= |volume=|issue= |pages= |doi= |arxiv=0801.0114}}, Приступљено 24. 4. 2013.</ref>

[[Ксиан-Јин Ли]] (-{Xian-Jin Li}-) у јулу 2008. године, неколико дана касније је повукао доказ због грешке (на страни 29).<ref>{{cite journal |author= |date= |title=Ксиан-Јин Ли (-{Xian-Jin Li}-), Доказ Риманове Хипотезе |url= |journal= |volume=|issue= |pages= |doi= |arxiv=0807.0090}}, Приступљено 24. 4. 2013.</ref>

[[Мајкл Атија]] је предложио доказ Риманове хипотезе 2018. године.<ref>[http://www.politika.rs/scc/clanak/411770/Britanac-resio-matematicki-problem-star-160-godina Британац решио математички проблем стар 160 година („Политика”, 24. септембар 2018)]</ref>

== Потрага за нулама Риманове зета-функције ==
Дуго се веровало да је Риманова хипотеза резултат дубоке интуиције и осећаја за проблем. Карл Лудвиг Сигел (-{Carl Ludwig Siegel}-) је, међутим, у тридесетим годинама 20. века анализирајући Риманове рукописе пронашао рачун за првих неколико нула на критичној правој, на неколико децималних цифара тачности.

Риманова хипотеза је нумерички проверена за првих 10<sup>13</sup> нула (за вредности ''-{t}-'' на критичној линији до 2,4&middot;10<sup>12</sup>). Овај резултат су 2004. године добили Ксавијер Гордон (-{Xavier Gourdon}-) и Патрик Демишел (-{Patrick Demichel}-) користећи Одлизко-Шонаге (-{Odlyzko-Schönhage}-) алгоритам<ref>{{cite journal|doi=10.1090/S0002-9947-1988-0961614-2|jstor=2000939 |title=Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function |last1=Odlyzko |first1=A. M. |last2=Schönhage |first2=A. |journal=Transactions of the American Mathematical Society |date=1988 |volume=309 |issue=2 |pages=797–809 }}</ref> из 1988. године.

Све познате вредности ''-{t}-'' за нуле на критичној линији су по свему судећи ирационални бројеви.

Све познате нуле су првог реда. Иако постојење нула вишег реда не би оповргло Риманову хипотезу - изазвало би озбиљне проблеме за доста савремених рачунских техника.

== Референце ==
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* {{Citation | last1=Rockmore | first1=Dan | title=Stalking the Riemann hypothesis | publisher=Pantheon Books | isbn=978-0-375-42136-5 | mr=2269393 | year=2005 | url=https://archive.org/details/stalkingriemannh00danr }}
* {{Citation | last1=Derbyshire | first1=John | author1-link=John Derbyshire | title=Prime Obsession | publisher=Joseph Henry Press, Washington, DC | isbn=978-0-309-08549-6 | mr=1968857 | year=2003| title-link=Prime Obsession }}
* {{Citation | last1=Watkins | first1=Matthew | author1-link=| title=Mystery of the Prime Numbers | publisher=Liberalis Books | isbn=978-1782797814 | mr=0000000 | year=2015| title-link=Mystery of the Prime Numbers }}
* [[Edward Frenkel|Frenkel, Edward]] (2014), [https://www.google.com/search?sxsrf=ACYBGNTYki6txmHlRlwZbknChP9TyCDuFA%3A1568142650416&source=hp&ei=OvV3Xe6hF8aE-gTn6qu4BA&q=numberphile+riemann+hypothesis&oq=numberphile+riemann&gs_l=psy-ab.1.0.0j0i22i30l2.2619.9717..11617...1.0..0.573.3549.0j16j3j5-1......0....1..gws-wiz.......0i131j35i39j0i10j0i22i10i30.tM-qtiL3ED8 The Riemann Hypothesis] [[Numberphile]], Mar 11, 2014 (video)
{{Литература крај}}

== Спољашње везе ==
{{Commons category|Riemann hypothesis}}
{{Литература|30em}}
* [http://www.claymath.org/millennium-problems Миленијумски проблеми Математичког Института Клеј (-{Clay Mathematics Institute Millennium Problems}-)], (2000), објава награде од милион долара за решење Миленијумских проблема.
* [http://www.claymath.org/sites/default/files/riemann1859-1a_1.jpg Фототипија рукописа Римановог рада из 1859. године] {{Wayback|url=http://www.claymath.org/sites/default/files/riemann1859-1a_1.jpg |date=20140818204420 }}, у којем се први пут помиње Риманова хипотеза.
* [http://mathworld.wolfram.com/RiemannHypothesis.html -{Mathworld}-], текст посвећен Римановој хипотези на сајту -{wolfram}-.
* -{[[American institute of mathematics]], [http://www.aimath.org/WWN/rh/ Riemann hypothesis]}-
* -{Zeroes [https://www.lmfdb.org/zeros/zeta/ database], 103 800 788 359 zeroes}-
* -{[https://www.youtube.com/watch?v=VTveQ1ndH1c ''The Key to the Riemann Hypothesis'' - Numberphile], a [[YouTube]] video about the Riemann hypothesis by [[Numberphile]]}-
* {{citation|first=Tom|last=Apostol|authorlink=Tom M. Apostol|url=http://www.math.wisc.edu/~robbin/funnysongs.html#Zeta|title=Where are the zeros of zeta of s?}} -{Poem about the Riemann hypothesis, [http://www.olimu.com/RIEMANN/Song.htm sung] by [[John Derbyshire]].}-
* {{citation|title=The Riemann Hypothesis|first=Peter|last=Borwein|authorlink=Peter Borwein|url=http://oldweb.cecm.sfu.ca/~pborwein/COURSE/MATH08/LECTURE.pdf|url-status=dead|archiveurl=https://web.archive.org/web/20090327181245/http://oldweb.cecm.sfu.ca/~pborwein/COURSE/MATH08/LECTURE.pdf|archivedate=2009-03-27}} 
* {{Citation | last1=Conrad | first1=K. | title=Consequences of the Riemann hypothesis | url=https://mathoverflow.net/q/17232 | year=2010}}
* {{citation|url=http://aimath.org/pl/rhequivalences|first=J. Brian|last=Conrey|first2=David W|last2=Farmer|title=Equivalences to the Riemann hypothesis|url-status=dead|archiveurl=https://web.archive.org/web/20100316235054/http://aimath.org/pl/rhequivalences|archivedate=2010-03-16}}
* {{citation|first= Xavier|last= Gourdon |first2= Pascal |last2=Sebah |url=http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html |title=Computation of zeros of the Zeta function|year=2004}}
* {{citation|last=Odlyzko|first=Andrew|authorlink=Andrew Odlyzko|url=http://www.dtc.umn.edu/~odlyzko/|title=Home page}} -{including [http://www.dtc.umn.edu/~odlyzko/doc/zeta.html papers on the zeros of the zeta function] and [http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html tables of the zeros of the zeta function]}-
* {{citation|last=Odlyzko|first=Andrew|authorlink=Andrew Odlyzko|title=Zeros of the Riemann zeta function: Conjectures and computations|year=2002|url=http://www.dtc.umn.edu/~odlyzko/talks/riemann-conjectures.pdf}} Slides of a talk
* {{citation|authorlink=Ed Pegg, Jr.|first= Ed |last=Pegg|url=http://www.maa.org/editorial/mathgames/mathgames_10_18_04.html |title=Ten Trillion Zeta Zeros|year=2004|publisher=Math Games website}}.
* {{citation|first=Glen|last= Pugh|url=http://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html |title=Java applet for plotting Z(t)}}
* {{citation |first=Michael |last=Rubinstein |url=http://pmmac03.math.uwaterloo.ca/~mrubinst/l_function_public/L.html |title=algorithm for generating the zeros |url-status=dead |archiveurl=https://web.archive.org/web/20070427221654/http://pmmac03.math.uwaterloo.ca/~mrubinst/l_function_public/L.html |archivedate=2007-04-27 }}.
* {{citation|first=Marcus|last=du Sautoy|authorlink=Marcus du Sautoy|url=http://www.seedmagazine.com/news/2006/03/prime_numbers_get_hitched.php|title=Prime Numbers Get Hitched|publisher=Seed Magazine|year=2006|access-date=2006-03-27|archive-url=https://web.archive.org/web/20170922145127/http://seedmagazine.com/content/article/prime_numbers_get_hitched/|archive-date=2017-09-22|url-status=dead}}
* {{citation|last=Stein|first=William A.|authorlink=William A. Stein|url=http://modular.math.washington.edu/edu/2007/simuw07/index.html|title=What is Riemann's hypothesis|url-status=dead|archiveurl=https://web.archive.org/web/20090104104251/http://modular.math.washington.edu/edu/2007/simuw07/index.html|archivedate=2009-01-04}}
* {{citation|last=de Vries|first=Andreas|url=http://math-it.org/Mathematik/Riemann/RiemannApplet.html|title=The Graph of the Riemann Zeta function ζ(s)|year=2004}}, a simple animated Java applet.
* {{citation | first=Matthew R. |last=Watkins|url=http://secamlocal.ex.ac.uk/~mwatkins/zeta/RHproofs.htm | title=Proposed proofs of the Riemann Hypothesis | date=2007-07-18}}
* -{''[https://web.archive.org/web/20131005173705/http://www.zetagrid.net/ Zetagrid]'' (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005}-
{{Литература крај}}

{{нормативна контрола}}

[[Категорија:Зета и L-функције]]
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