Riemannova domneva: Razlika med redakcijama

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=== Kvazikristali ===
 
Iz Riemannove domneve izhaja, da ničle funkcije ζ tvorijo [[kvazikristal]], kar pomeni porazdelitev nezvezne opore, katere [[FourierjevaFourierova transformacija]] ima tudi nezvezno oporo. [[Freeman John Dyson|Dyson]]<ref>{{sktxt|Dyson|2009}}.</ref> je leta 2009 predlagal poskus dokaza Riemannove domneve s klasifikacijo, oziroma vsaj z raziskovanjem 1-razsežnih kvazikristalov, ki imajo dosti bogatejšo vsebino od na primer trirazsežnih. Nekateri menijo, da njegova definicija kvazikristalov ni ustrezna.
 
=== Aritmetične funkcije ζ modelov eliptičnih krivulj številskih obsegov ===
== Viri ==
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* {{citat|last1= Odlyzko|first1= Andrew Michael|authorlink1= |title= The 10<sup>20</sup>-th zero of the Riemann zeta function and 175 million of its neighbors|yeardate= 1992|url=http://www.dtc.umn.edu/~odlyzko/unpublished/zeta.10to20.1992.pdf|ref= harv}} Ta neobjavljena knjiga opisuje implementacijo algoritma in podrobno obravnava rezultate.
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* {{citat|last1= Selberg|first1= Atle|authorlink1= |title= Contributions to the theory of the Riemann zeta-function|mr= 0020594|yeardate= 1946|journal= Arch. Math. Naturvid.|volume= 48|issue= 5|pages= 89–155|ref= harv}}
* {{citat|last1= Selberg|first1= Atle|authorlink1= |title= Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series|mr= 0088511|yeardate= 1956|journal= J. Indian Math. Soc. (N.S.)|volume= 20|pages= 47–87|ref= harv}}
* {{citat|last1= Serre|first1= Jean-Pierre|authorlink1= Jean-Pierre Serre|title= Facteurs locaux des fonctions zeta des varietés algébriques (définitions et conjectures)|journal= Séminaire Delange-Pisot-Poitou|yeardate= 1969–1970|volume= 19|url= https://eudml.org/doc/110758|ref= harv}}
* {{citat|last1= Sheats|first1= Jeffrey T.|title= The Riemann hypothesis for the Goss zeta function for '''F'''<sub>q</sub>[T]|doi= 10.1006/jnth.1998.2232|mr= 1630979|journal= [[Journal of Number Theory|J. Number Theory]]|date= 1998|volume= 71|issue= 1|pages= 121–157|ref= harv}}
* {{citat|last1= Siegel|first1= Carl Ludwig|authorlink1= Carl Ludwig Siegel|title= Über Riemanns Nachlaß zur analytischen Zahlentheorie|journal= Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2|pages= 45–80|yeardate= 1932|ref= harv}} Ponatisnjeno v Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966.
* {{citat|last1= Speiser|first1= Andreas|authorlink1= Andreas Speiser|title= Geometrisches zur Riemannschen Zetafunktion|yeardate= 1934|journal= [[Mathematische Annalen]]|volume= 110|pages= 514–521|doi= 10.1007/BF01448042|jfm= 60.0272.04|url= http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0110&DMDID=DMDLOG_0032&L=1|ref= harv}}
* {{citat|last1= Spira|first1= Robert|journal= [[Mathematics of Computation]]|mr= 0228456|pages= 163–173|title= Zeros of sections of the zeta function. II|volume= 22|yeardate= 1968|doi= 10.2307/2004774|ref= harv}}
* {{citat|last1= Suzuki|first1= Masatoši|title= Positivity of certain functions associated with analysis on elliptic surfaces|journal= [[Journal of Number Theory|J. Number Theory]]|date= 2011|volume= 131|issue= 10|pages= 1770–1796|doi= 10.1016/j.jnt.2011.03.007|ref= harv}}
* {{citat|last1= Titchmarsh|first1= Edward Charles|authorlink1= Edward Charles Titchmarsh|title= The Zeros of the Riemann Zeta-Function|jstor= 96545|publisher= Kraljeva družba|yeardate= 1935|journal= Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|volume= 151|issue= 873|pages= 234–255|doi= 10.1098/rspa.1935.0146|ref= harv}}
* {{citat|last1= Titchmarsh|first1= Edward Charles|authorlink1= |title= The Zeros of the Riemann Zeta-Function|jstor= 96692|publisher= Kraljeva družba|yeardate= 1936|journal= Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences|volume= 157|issue= 891|pages= 261–263|doi= 10.1098/rspa.1936.0192|ref= harv}}
* {{citat|last1= Titchmarsh|first1= Edward Charles|authorlink1= |title= The theory of the Riemann zeta-function|publisher= The Clarendon Press Oxford University Press|edition= 2.|isbn= 978-0-19-853369-6|mr= 882550|yeardate= 1986|ref= harv}}
* {{citat|last1= Trudgian|first1= Timothy|title= On the success and failure of Gram's Law and the Rosser Rule|yeardate= 2011|journal= Acta Arithmetica|volume= 125|issue= 3|pages= 225–256|doi= 10.4064/aa148-3-2|ref= harv}}
* {{citat|last1= Turán|first1= Paul|authorlink1= Pál Turán|title= On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann|mr= 0027305|yeardate= 1948|journal= Danske Vid. Selsk. Mat.-Fys. Medd.|volume= 24|issue= 17|pages= 36|ref= harv}} Ponatisnjeno v {{harv|Borwein|Choi|Rooney|Weirathmueller|2008}}.
* {{citat|last1= Turing|first1= Alan Mathison|authorlink1= Alan Turing|title= Some calculations of the Riemann zeta-function|doi= 10.1112/plms/s3-3.1.99|mr= 0055785|yeardate= 1953|journal= [[Proceedings of the London Mathematical Society|Proc. London Math. Soc.]]. Third Series|volume= 3|issue= 1|pages= 99–117|ref= harv}}
* {{citat|last1= La Vallée Poussin|first1= Charles-Jean de|authorlink1= Charles-Jean de La Vallée Poussin|title= Recherches analytiques sur la théorie des nombers premiers|journal= Ann. Soc. Sci. Bruxelles|volume= 20|yeardate= 1896|pages= 183–256|ref= harv}}
* {{citat|last1= La Vallée Poussin|first1= Charles-Jean de|authorlink1= |title= Sur la fonction ζ(s) de Riemann et la nombre des nombres premiers inférieurs à une limite donnée|journal= Mem. Couronnes Acad. Sci. Belg.|volume= 59|issue= 1|yeardate= 1899–1900|ref= harv}} Ponatisnjeno v {{harv|Borwein|Choi|Rooney|Weirathmueller|2008}}.
* {{citat|last1= Volčkov|first1= V. V.|title= On an equality equivalent to the Riemann hypothesis|journal= Ukrainian Mathematical Journal|date= 1995-03|volume= 47|issue= 3|pages= 491–493|doi= 10.1007/BF01056314|ref= harv}}
* {{citat|last1= Weil|first1= André|authorlink1= André Weil|title= Sur les courbes algébriques et les variétés qui s'en déduisent|publisher= Hermann et Cie., Paris|series= Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945)|mr= 0027151|yeardate= 1948|ref= harv}}
* {{citat|last1= Weil|first1= André|authorlink1= |title= Numbers of solutions of equations in finite fields|doi= 10.1090/S0002-9904-1949-09219-4|mr= 0029393|yeardate= 1949|journal= [[Bulletin of the American Mathematical Society]]|volume= 55|pages= 497–508|issue= 5|ref= harv}} Ponatisnjeno v in Oeuvres Scientifiques/Collected Papers by Andre Weil ISBN 0-387-90330-5
* {{citat|last1= Weinberger|first1= Peter Jay|authorlink1= Peter Jay Weinberger|title= Analytic number theory ( St. Louis Univ., 1972)|publisher= Ameriško matematično društvo|location= Providence, R.I.|series= Proc. Sympos. Pure Math.|mr= 0337902|yeardate= 1973|volume= 24|chapter= On Euclidean rings of algebraic integers|pages= 321–332|ref= harv}}
* {{citat|last1= Wiles|first1= Andrew|authorlink1= Andrew Wiles|title= Mathematics: frontiers and perspectives|publisher= Ameriško matematično društvo|location= Providence, R.I.|isbn= 978-0-8218-2697-3|mr= 1754786|yeardate= 2000|chapter= Twenty years of number theory|pages= 329–342|ref= harv}}
* {{navedi arXiv|last1= Wolf|first1= Marek|authorlink1= Marek Wolf|title= Failed attempt to disproof the Riemann Hypothesis|date= 2009-10-08|arxiv= 0910.1534|class= math.NT|ref= harv}}
* {{navedi arXiv|last1= Wolf|first1= Marek|authorlink1= |title= Will a physicists prove the Riemann Hypothesis?|date= 2014-10-09|arxiv= 1410.1214|class= math-ph|ref= harv}}
* {{citat|last1= Zagier|first1= Don|authorlink1 = Don Zagier|url= http://modular.math.washington.edu/edu/2007/simuw07/misc/zagier-the_first_50_million_prime_numbers.pdf|format= PDF|publisher= Springer|location= |mr= 643810|yeardate= 1977|volume= 0|title= The first 50 million prime numbers|pages= 7–19|journal= Math. Intelligencer|doi= 10.1007/BF03039306|ref= harv}}
* {{citat|last1= Zagier|first1= Don|authorlink1= |title= Automorphic forms, representation theory and arithmetic (Bombay, 1979)|publisher= Tata Inst. Fundamental Res., Bombay|series= Tata Inst. Fund. Res. Studies in Math.|mr= 633666|yeardate= 1981|volume= 10|chapter= Eisenstein series and the Riemann zeta function|pages= 275–301|ref= harv}}
{{refend}}