Hipoteza (matematika)

Realni deo (crveno) i imaginarni deo (plavo) Rimanove zeta funkcije duž kritične linije Re(s) = 1/2. Prve netrivijalne nule mogu se videti kod Im(s) = ±14,135, ±21,022 i ±25,011. Rimanova hipoteza je poznata hipoteza, kaže da sve netrivijalne nule zeta funkcije leže duž kritične linije.

U matematici, hipoteza je zaključak ili predlog za koji se pretpostavlja da je istinit zbog preliminarnih pratećih dokaza, ali za koji još nije pronađen dokaz ili opovrgnuće.^[1]^[2]^[3]^[4] Neke konjekture, kao što je Rimanova hipoteza (još uvek pretpostavka) ili poslednja Fermaova teorema (pretpostavka koju je Endru Vajls dokazao 1995. godine), oblikovali su veći deo matematičke istorije, jer su razvijena nova područja matematike kako bi se dokazale.^[5]

Važni primeri

Poslednja Fermaova teorema

U teoriji brojeva, Poslednja Fermaova teorema (ponekad zvana Fermaova konjektura, posebno u starijim tekstovima) navodi da nijedna tri pozitivna cela broja $a$ , $b$ , i $c$ ne mogu da zadovolje jednačinu $a^{n}+b^{n}=c^{n}$ za bilo koju celobrojnu vrednost $n$ veću od dva.

Ovu teoremu je prvi formulisao Pjer de Ferma 1637. godine na marginama kopije Diofantove Aritmetike, gde je tvrdio da ima dokaz koji je prevelik da bi stao u marginu.^[6] Prvi uspešan dokaz objavio je Endru Vajls 1994. godine, a formalno je objavljen 1995. godine, nakon 358 godina napora matematičara. Ovaj nerešeni problem podstakao je razvoj algebarske teorije brojeva u 19. veku, i dokaz teoreme modularnosti u 20. veku. Ovo je jedna je od najistaknutijih teorema u istoriji matematike, a pre njenog dokaza bila je u Ginisovoj knjizi svetskih rekorda kao jedan od „najtežih matematičkih problema”.^[7]

Reference

^ „The Definitive Glossary of Higher Mathematical Jargon — Conjecture”. Math Vault (на језику: енглески). 2019-08-01. Приступљено 2019-11-12.
^ „Definition of CONJECTURE”. www.merriam-webster.com (на језику: енглески). Приступљено 2019-11-12.
^ Oxford Dictionary of English (2010 изд.).
^ Schwartz, JL (1995). Shuttling between the particular and the general: reflections on the role of conjecture and hypothesis in the generation of knowledge in science and mathematics. стр. 93. ISBN 9780195115772.
^ Weisstein, Eric W. „Fermat's Last Theorem”. mathworld.wolfram.com (на језику: енглески). Приступљено 2019-11-12.
^ Ore, Oystein (1988) [1948], Number Theory and Its History, Dover, стр. 203–204, ISBN 978-0-486-65620-5
^ „Science and Technology”. The Guinness Book of World Records. Guinness Publishing Ltd. 1995.

Literatura

Beurling, Arne (1955), „A closure problem related to the Riemann zeta-function”, Proceedings of the National Academy of Sciences of the United States of America, 41 (5): 312—314, Bibcode:1955PNAS...41..312B, MR 0070655, PMC 528084 , PMID 16589670, doi:10.1073/pnas.41.5.312
Bombieri, Enrico (2000), The Riemann Hypothesis – official problem description (PDF), Clay Mathematics Institute, Архивирано из оригинала (PDF) 22. 12. 2015. г., Приступљено 2008-10-25 Reprinted in Borwein et al. 2008.
Borwein, Peter; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea, ур. (2008), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, New York: Springer, ISBN 978-0-387-72125-5, doi:10.1007/978-0-387-72126-2
Borwein, Peter; Ferguson, Ron; Mossinghoff, Michael J. (2008), „Sign changes in sums of the Liouville function”, Mathematics of Computation, 77 (263): 1681—1694, Bibcode:2008MaCom..77.1681B, MR 2398787, doi:10.1090/S0025-5718-08-02036-X
de Branges, Louis (1992), „The convergence of Euler products”, Journal of Functional Analysis, 107 (1): 122—210, MR 1165869, doi:10.1016/0022-1236(92)90103-P
Broughan, Kevin (2017), Equivalents of the Riemann Hypothesis, Cambridge University Press, ISBN 978-1108290784
Burton, David M. (2006), Elementary Number Theory, Tata McGraw-Hill Publishing Company Limited, ISBN 978-0-07-061607-3
Cartier, P. (1982), „Comment l'hypothèse de Riemann ne fut pas prouvée”, Seminar on Number Theory, Paris 1980–81 (Paris, 1980/1981), Progr. Math., 22, Boston, MA: Birkhäuser Boston, стр. 35—48, MR 693308
Connes, Alain (1999), „Trace formula in noncommutative geometry and the zeros of the Riemann zeta function”, Selecta Mathematica. New Series, 5 (1): 29—106, MR 1694895, arXiv:math/9811068 , doi:10.1007/s000290050042
Connes, Alain (2000), „Noncommutative geometry and the Riemann zeta function”, Mathematics: frontiers and perspectives, Providence, R.I.: American Mathematical Society, стр. 35—54, MR 1754766
Connes, Alain (2016), „An Essay on the Riemann Hypothesis”, Ур.: Nash, J. F.; Rassias, Michael, Open Problems in Mathematics, New York: Springer, стр. 225—257, arXiv:1509.05576 , doi:10.1007/978-3-319-32162-2_5
Conrey, J. B. (1989), „More than two fifths of the zeros of the Riemann zeta function are on the critical line”, J. Reine Angew. Math., 1989 (399): 1—16, MR 1004130, doi:10.1515/crll.1989.399.1, Архивирано из оригинала 26. 07. 2023. г., Приступљено 09. 03. 2020
Conrey, J. Brian (2003), „The Riemann Hypothesis” (PDF), Notices of the American Mathematical Society: 341—353 Reprinted in Borwein et al. 2008.
Conrey, J. B.; Li, Xian-Jin (2000), „A note on some positivity conditions related to zeta and L-functions”, International Mathematics Research Notices, 2000 (18): 929—940, MR 1792282, arXiv:math/9812166 , doi:10.1155/S1073792800000489
Deligne, Pierre (1974), „La conjecture de Weil. I”, Publications Mathématiques de l'IHÉS, 43: 273—307, MR 0340258, doi:10.1007/BF02684373
Deligne, Pierre (1980), „La conjecture de Weil : II”, Publications Mathématiques de l'IHÉS, 52: 137—252, doi:10.1007/BF02684780
Deninger, Christopher (1998), „Some analogies between number theory and dynamical systems on foliated spaces”, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Documenta Mathematica, стр. 163—186, MR 1648030
Dudek, Adrian W. (2014-08-21), „On the Riemann hypothesis and the difference between primes”, International Journal of Number Theory, 11 (3): 771—778, Bibcode:2014arXiv1402.6417D, ISSN 1793-0421, arXiv:1402.6417 , doi:10.1142/S1793042115500426
Dyson, Freeman (2009), „Birds and frogs” (PDF), Notices of the American Mathematical Society, 56 (2): 212—223, MR 2483565
Edwards, H. M. (1974), Riemann's Zeta Function, New York: Dover Publications, ISBN 978-0-486-41740-0, MR 0466039
Fesenko, Ivan (2010), „Analysis on arithmetic schemes. II”, Journal of K-theory, 5 (3): 437—557, doi:10.1017/is010004028jkt103
Ford, Kevin (2002), „Vinogradov's integral and bounds for the Riemann zeta function”, Proceedings of the London Mathematical Society, Third Series, 85 (3): 565—633, MR 1936814, doi:10.1112/S0024611502013655
Franel, J.; Landau, E. (1924), „Les suites de Farey et le problème des nombres premiers" (Franel, 198–201); "Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel (Landau, 202–206)”, Göttinger Nachrichten: 198—206
Ghosh, Amit (1983), „On the Riemann zeta function—mean value theorems and the distribution of |S(T)|”, J. Number Theory, 17: 93—102, doi:10.1016/0022-314X(83)90010-0
Gourdon, Xavier (2004), The 10¹³ first zeros of the Riemann Zeta function, and zeros computation at very large height (PDF)
Gram, J. P. (1903), „Note sur les zéros de la fonction ζ(s) de Riemann” (PDF), Acta Mathematica, 27: 289—304, doi:10.1007/BF02421310
Hadamard, Jacques (1896), „Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques”, Bulletin de la Société Mathématique de France, 14: 199—220, doi:10.24033/bsmf.545 Reprinted in Borwein et al. 2008.
Hardy, G. H. (1914), „Sur les Zéros de la Fonction ζ(s) de Riemann”, C. R. Acad. Sci. Paris, 158: 1012—1014, JFM 45.0716.04 Reprinted in Borwein et al. 2008.
Hardy, G. H.; Littlewood, J. E. (1921), „The zeros of Riemann's zeta-function on the critical line”, Math. Z., 10 (3–4): 283—317, doi:10.1007/BF01211614
Haselgrove, C. B. (1958), „A disproof of a conjecture of Pólya”, Mathematika, 5 (2): 141—145, ISSN 0025-5793, MR 0104638, Zbl 0085.27102, doi:10.1112/S0025579300001480 Reprinted in Borwein et al. 2008.
Haselgrove, C. B.; Miller, J. C. P. (1960), Tables of the Riemann zeta function, Royal Society Mathematical Tables, Vol. 6, Cambridge University Press, ISBN 978-0-521-06152-0, MR 0117905 Review
Hutchinson, J. I. (1925), „On the Roots of the Riemann Zeta-Function”, Transactions of the American Mathematical Society, 27 (1): 49—60, JSTOR 1989163, doi:10.2307/1989163
Ingham, A.E. (1932), The Distribution of Prime Numbers, Cambridge Tracts in Mathematics and Mathematical Physics, 30, Cambridge University Press . Reprinted 1990, ISBN 978-0-521-39789-6, MR1074573
Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X
Ivić, A. (1985), The Riemann Zeta Function, New York: John Wiley & Sons, ISBN 978-0-471-80634-9, MR 0792089 (Reprinted by Dover 2003)
Ivić, Aleksandar (2008), „On some reasons for doubting the Riemann hypothesis”, Ур.: Borwein, Peter; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea, The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, New York: Springer, стр. 131—160, ISBN 978-0-387-72125-5, arXiv:math.NT/0311162 
Karatsuba, A. A. (1984a), „Zeros of the function ζ(s) on short intervals of the critical line”, Izv. Akad. Nauk SSSR, Ser. Mat. (на језику: руски), 48 (3): 569—584, MR 0747251
Karatsuba, A. A. (1984b), „Distribution of zeros of the function ζ(1/2 + it)”, Izv. Akad. Nauk SSSR, Ser. Mat. (на језику: руски), 48 (6): 1214—1224, MR 0772113
Karatsuba, A. A. (1985), „Zeros of the Riemann zeta-function on the critical line”, Trudy Mat. Inst. Steklov. (на језику: руски) (167): 167—178, MR 0804073
Karatsuba, A. A. (1992), „On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line”, Izv. Ross. Akad. Nauk, Ser. Mat. (на језику: руски), 56 (2): 372—397, Bibcode:1993IzMat..40..353K, MR 1180378, doi:10.1070/IM1993v040n02ABEH002168
Karatsuba, A. A.; Voronin, S. M. (1992), The Riemann zeta-function, de Gruyter Expositions in Mathematics, 5, Berlin: Walter de Gruyter & Co., ISBN 978-3-11-013170-3, MR 1183467, doi:10.1515/9783110886146
Keating, Jonathan P.; Snaith, N. C. (2000), „Random matrix theory and ζ(1/2 + it)”, Communications in Mathematical Physics, 214 (1): 57—89, Bibcode:2000CMaPh.214...57K, MR 1794265, doi:10.1007/s002200000261
Knauf, Andreas (1999), „Number theory, dynamical systems and statistical mechanics”, Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics, 11 (8): 1027—1060, Bibcode:1999RvMaP..11.1027K, MR 1714352, doi:10.1142/S0129055X99000325
von Koch, Niels Helge (1901), „Sur la distribution des nombres premiers”, Acta Mathematica, 24: 159—182, doi:10.1007/BF02403071
Kurokawa, Nobushige (1992), „Multiple zeta functions: an example”, Zeta functions in geometry (Tokyo, 1990), Adv. Stud. Pure Math., 21, Tokyo: Kinokuniya, стр. 219—226, MR 1210791
Lapidus, Michel L. (2008), In search of the Riemann zeros, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4222-5, MR 2375028, doi:10.1090/mbk/051
Шаблон:Eom
Lehmer, D. H. (1956), „Extended computation of the Riemann zeta-function”, Mathematika. A Journal of Pure and Applied Mathematics, 3 (2): 102—108, MR 0086083, doi:10.1112/S0025579300001753
Leichtnam, Eric (2005), „An invitation to Deninger's work on arithmetic zeta functions”, Geometry, spectral theory, groups, and dynamics, Contemp. Math., 387, Providence, RI: Amer. Math. Soc., стр. 201—236, MR 2180209, doi:10.1090/conm/387/07243 .
Levinson, N. (1974), „More than one-third of the zeros of Riemann's zeta function are on σ = 1/2”, Adv. Math., 13 (4): 383—436, MR 0564081, doi:10.1016/0001-8708(74)90074-7
Littlewood, J. E. (1962), „The Riemann hypothesis”, The scientist speculates: an anthology of partly baked idea, New York: Basic books
van de Lune, J.; te Riele, H. J. J.; Winter, D. T. (1986), „On the zeros of the Riemann zeta function in the critical strip. IV”, Mathematics of Computation, 46 (174): 667—681, JSTOR 2008005, MR 829637, doi:10.2307/2008005
Massias, J.-P.; Nicolas, Jean-Louis; Robin, G. (1988), „Évaluation asymptotique de l'ordre maximum d'un élément du groupe symétrique”, Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica, 50 (3): 221—242, MR 960551, doi:10.4064/aa-50-3-221-242, Архивирано из оригинала 16. 03. 2012. г., Приступљено 09. 03. 2020
Mazur, Barry; Stein, William (2015), Prime Numbers and the Riemann Hypothesis
Montgomery, Hugh L. (1973), „The pair correlation of zeros of the zeta function”, Analytic number theory, Proc. Sympos. Pure Math., XXIV, Providence, R.I.: American Mathematical Society, стр. 181—193, MR 0337821 Reprinted in Borwein et al. 2008.
Montgomery, Hugh L. (1983), „Zeros of approximations to the zeta function”, Ур.: Erdős, Paul, Studies in pure mathematics. To the memory of Paul Turán, Basel, Boston, Berlin: Birkhäuser, стр. 497—506, ISBN 978-3-7643-1288-6, MR 820245
Montgomery, Hugh L.; Vaughan, Robert C. (2007), Multiplicative Number Theory I. Classical Theory, Cambridge studies in advanced mathematics, 97, Cambridge University Press .ISBN 978-0-521-84903-6
Nicely, Thomas R. (1999), „New maximal prime gaps and first occurrences”, Mathematics of Computation, 68 (227): 1311—1315, Bibcode:1999MaCom..68.1311N, MR 1627813, doi:10.1090/S0025-5718-99-01065-0, Архивирано из оригинала 30. 12. 2014. г., Приступљено 09. 03. 2020 .
Nyman, Bertil (1950), On the One-Dimensional Translation Group and Semi-Group in Certain Function Spaces, PhD Thesis, University of Uppsala: University of Uppsala, MR 0036444