Rimanova hipoteza

Rimanova hipoteza je pretpostavka o distribuciji netrivijalnih nula Rimanove zeta-funkcije $\zeta (s)\,$. Prvi put je formulisana u radu Bernarda Rimana iz 1859: O broju prostih brojeva ispod zadate veličine (nem. Über der Anzahl der Primzahlen unter einer gegebenen Größe). Od tada, i pored ogromnih napora, ovaj problem i dalje ostaje nerešen.

Realni deo (u crvenoj) i imaginarni deo (u plavoj boji) Rimanove zeta-funkcije na kritičnoj liniji Re(s) = 1/2. Prvih nekoliko imaginarnih nula se može videti za vrednosti Im(s) = ±14,135, ±21,022 i ±25,011.

Rimanova zeta-funkcija je definisana za sve kompleksne brojeve s ≠ 1, i ima trivijalne nule u parnim negativnim celim brojevima (s = −2, s = −4, s = −6, ...). Rimanova hipoteza kaže da se sve netrivijalne nule nalaze na jednoj pravoj u kompleksnoj ravni, konkretno:

Realni deo bilo koje netrivijalne nule Rimanove zeta-funkcije je ½, odnosno sve netrivijalne nule se nalaze na kritičnoj liniji ½ + it.

Istorijat

Rad iz 1859. je Rimanov jedini ogled u teoriji brojeva, ali je hipoteza izneta u njemu jedan od najznačajnijih nerešenih problema u savremenoj matematici, pre svega zato što se dosta važnih rezultata oslanja na važenje ove hipoteze (recimo u kriptografiji, faktorizaciji celih brojeva i polinoma).

Legenda kaže da se kopija sakupljenih Rimanovih radova u Hurvicovoj (engl. Adolf Hurwitz) biblioteci nakon njegove smrti sama otvarala na strani na kojoj se nalazio iskaz Rimanove hipoteze.

David Hilbert je na Drugom međunarodnom kongresu matematičara u Parizu, 8. avgusta 1900. godine postavio problem Rimanove hipoteze kao jedan od dvadesettri Hilbertova problema (problem broj osam). Za Hilberta je Rimanova hipoteza imala poseban značaj, kada su ga pitali šta bi najpre uradio nakon 500-godišnjeg sna, Hilbert je odgovorio da bi prvo pitao da li je Rimanova hipoteza dokazana.

Godfri Harold Hardi (engl. Godfrey Harold Hardy) je 1914. godine dokazao da se na kritičnoj liniji ½ + it nalazi beskonačno mnogo nula.

Rimanova hipoteza je jedan od sedam Milenijumskih problema Matematičkog instituta Klej.

Pokušaji dokazivanja

Rimanova hipoteza je kao i Poslednja Fermaova teorema bila inspiracija za nebrojene pokušaje dokazivanja, gde su podjednako neuspešni bili i vrhunski i matematičari amateri. Kada je 1995. godine engleski matematičar Endru Vajls izveo dokaz Fermaove poslednje teoreme - fokus matematičke zajednice je preusmeren na Rimanovu hipotezu, najistaknutiji nerešeni problem u matematici danas. Ovde su nabrojani značajni neuspešni pokušaji u novom milenijumu.

Mati Pitkanen (Matti Pitkanen) u septembru 2001, povukao dokaz zbog greške u novembru iste godine.

Karlos Kastro (Carlos Castro), i Horge Maheha (Jorge Mahecha) su u seriji radova od 2001. do 2006. godine probali da izgrade teoriju (koristeći supersimetrije i kvantnomehanički pristup) koja bi omogućila dokazivanje Rimanove hipoteze. Njihov pristup je odbačen.

Kaida Ši (Kaida Shi) u julu 2003. godine, dokaz sadržavao grešku.

Luj d'Branž (Louis de Branges de Bourcia) u julu 2004. godine, nađen kontraprimer. Autor je kasnije objavio Izvinjenje za dokaz Rimanove Hipoteze.

Jinžu Han (Jinzhu Han) u junu 2007. godine, dokaz sadržavao grešku.

Andrej Madrecki (Andrzej Madrecki) u julu 2007. godine, dokaz sadržavao grešku.

Lev Aizenberg (Lev Aizenberg) u decembru 2007. godine, povukao dokaz zbog greške u januaru 2008. godine.

Ksian-Jin Li (Xian-Jin Li) u julu 2008. godine, nekoliko dana kasnije je povukao dokaz zbog greške (na strani 29).

Majkl Atija je predložio dokaz Rimanove hipoteze 2018. godine.

Potraga za nulama Rimanove zeta-funkcije

Dugo se verovalo da je Rimanova hipoteza rezultat duboke intuicije i osećaja za problem. Karl Ludvig Sigel (Carl Ludwig Siegel) je, međutim, u tridesetim godinama 20. veka analizirajući Rimanove rukopise pronašao račun za prvih nekoliko nula na kritičnoj pravoj, na nekoliko decimalnih cifara tačnosti.

Rimanova hipoteza je numerički proverena za prvih 10¹³ nula (za vrednosti t na kritičnoj liniji do 2,4·10¹²). Ovaj rezultat su 2004. godine dobili Ksavijer Gordon (Xavier Gourdon) i Patrik Demišel (Patrick Demichel) koristeći Odlizko-Šonage (Odlyzko-Schönhage) algoritam iz 1988. godine.

Sve poznate vrednosti t za nule na kritičnoj liniji su po svemu sudeći iracionalni brojevi.

Sve poznate nule su prvog reda. Iako postojenje nula višeg reda ne bi opovrglo Rimanovu hipotezu - izazvalo bi ozbiljne probleme za dosta savremenih računskih tehnika.

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• Turán, Paul (1948), „On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann”, Danske Vid. Selsk. Mat.-Fys. Medd., 24 (17): 36, MR 0027305  Reprinted in Borwein et al. 2008.
• Turing, Alan M. (1953), „Some calculations of the Riemann zeta-function”, Proceedings of the London Mathematical Society, Third Series, 3: 99—117, MR 0055785, doi:10.1112/plms/s3-3.1.99 
• de la Vallée-Poussin, Ch.J. (1896), „Recherches analytiques sur la théorie des nombers premiers”, Ann. Soc. Sci. Bruxelles, 20: 183—256 
• de la Vallée-Poussin, Ch.J. (1899—1900), „Sur la fonction ζ(s) de Riemann et la nombre des nombres premiers inférieurs à une limite donnée”, Mem. Couronnes Acad. Sci. Belg., 59 (1)  Reprinted in Borwein et al. 2008.
• Weil, André (1948), Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, MR 0027151 
• Weil, André (1949), „Numbers of solutions of equations in finite fields”, Bulletin of the American Mathematical Society, 55 (5): 497—508, MR 0029393, doi:10.1090/S0002-9904-1949-09219-4   Reprinted in Oeuvres Scientifiques/Collected Papers by Andre Weil ISBN 0-387-90330-5
• Weinberger, Peter J. (1973), „On Euclidean rings of algebraic integers”, Analytic number theory ( St. Louis Univ., 1972), Proc. Sympos. Pure Math., 24, Providence, R.I.: Amer. Math. Soc., str. 321—332, MR 0337902 
• Wiles, Andrew (2000), „Twenty years of number theory”, Mathematics: frontiers and perspectives, Providence, R.I.: American Mathematical Society, str. 329—342, ISBN 978-0-8218-2697-3, MR 1754786 
• Zagier, Don (1977), „The first 50 million prime numbers” (PDF), Math. Intelligencer, Springer, 0: 7—19, MR 643810, doi:10.1007/BF03039306, Arhivirano iz originala (PDF) 2009-03-27. g. 
• Zagier, Don (1981), „Eisenstein series and the Riemann zeta function”, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay, str. 275—301, MR 633666 
• Sabbagh, Karl (2003a), The greatest unsolved problem in mathematics, Farrar, Straus and Giroux, New York, ISBN 978-0-374-25007-2, MR 1979664 
• Sabbagh, Karl (2003b), Dr. Riemann's zeros, Atlantic Books, London, ISBN 978-1-843-54101-1 
• du Sautoy, Marcus (2003), The music of the primes, HarperCollins Publishers, ISBN 978-0-06-621070-4, MR 2060134 
• Rockmore, Dan (2005), Stalking the Riemann hypothesis, Pantheon Books, ISBN 978-0-375-42136-5, MR 2269393 
• Derbyshire, John (2003), Prime Obsession, Joseph Henry Press, Washington, DC, ISBN 978-0-309-08549-6, MR 1968857 
• Watkins, Matthew (2015), Mystery of the Prime Numbers, Liberalis Books, ISBN 978-1782797814, MR 0000000 
• Frenkel, Edward (2014), The Riemann Hypothesis Numberphile, Mar 11, 2014 (video)

Spoljašnje veze

• Milenijumski problemi Matematičkog Instituta Klej (Clay Mathematics Institute Millennium Problems), (2000), objava nagrade od milion dolara za rešenje Milenijumskih problema.
• Fototipija rukopisa Rimanovog rada iz 1859. godine Arhivirano na sajtu Wayback Machine (18. avgust 2014), u kojem se prvi put pominje Rimanova hipoteza.
• Mathworld, tekst posvećen Rimanovoj hipotezi na sajtu wolfram.
• American institute of mathematics, Riemann hypothesis
• Zeroes database, 103 800 788 359 zeroes
• The Key to the Riemann Hypothesis - Numberphile, a YouTube video about the Riemann hypothesis by Numberphile
• Apostol, Tom, Where are the zeros of zeta of s?  Poem about the Riemann hypothesis, sung by John Derbyshire.
• Borwein, Peter, The Riemann Hypothesis (PDF), Arhivirano iz originala (PDF) 2009-03-27. g. 
• Conrad, K. (2010), Consequences of the Riemann hypothesis 
• Conrey, J. Brian; Farmer, David W, Equivalences to the Riemann hypothesis, Arhivirano iz originala 2010-03-16. g. 
• Gourdon, Xavier; Sebah, Pascal (2004), Computation of zeros of the Zeta function 
• Odlyzko, Andrew, Home page  including papers on the zeros of the zeta function and tables of the zeros of the zeta function
• Odlyzko, Andrew (2002), Zeros of the Riemann zeta function: Conjectures and computations (PDF)  Slides of a talk
• Pegg, Ed (2004), Ten Trillion Zeta Zeros, Math Games website .
• Pugh, Glen, Java applet for plotting Z(t) 
• Rubinstein, Michael, algorithm for generating the zeros, Arhivirano iz originala 2007-04-27. g. .
• du Sautoy, Marcus (2006), Prime Numbers Get Hitched, Seed Magazine, Arhivirano iz originala 2017-09-22. g., Pristupljeno 2006-03-27 
• Stein, William A., What is Riemann's hypothesis, Arhivirano iz originala 2009-01-04. g. 
• de Vries, Andreas (2004), The Graph of the Riemann Zeta function ζ(s) , a simple animated Java applet.
• Watkins, Matthew R. (2007-07-18), Proposed proofs of the Riemann Hypothesis 
• Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005