The Riemann hypothesis: the great pending mayhematical challenge

• Autores: Pilar Bayer Isant
• Localización: Mètode Science Studies Journal: Annual Review, ISSN 2174-3487, ISSN-e 2174-9221, Nº. 8, 2018 (Ejemplar dedicado a: Making Science. A multitude of perspectives), págs. 34-41
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • The Riemann hypothesis is an unproven statement referring to the zeros of the Riemann zeta function. Bernhard Riemann calculated the first six non-trivial zeros of the function and observed that they were all on the same straight line. In a report published in 1859, Riemann stated that this might very well be a general fact. The Riemann hypothesis claims that all non-trivial zeros of the zeta function are on the the line x=1/2. The more than ten billion zeroes calculated to date, all of them lying on the critical line, coincide with Riemann’s suspicion, but no one has yet been able to prove that the zeta function does not have non-trivial zeroes outside of this line.

• Referencias bibliográficas
  • Bayer, P. (2006). La hipòtesi de Riemann. In J. Quer (Ed.), Els set problemes del mil·lenni (pp. 29–62). Sabadell: Fundació Caixa Sabadell.
  • Bayer, P., & Neukirch, J. (1978). On values of zeta functions and ℓ-adic Euler characteristics. Inventiones Mathematicae, 50(1), 35–64....
  • Berry, M. V., & Keating, J. P. (1999). The Riemann zeros and eigenvalue asymptotics. SIAM Review, 41(2), 236–266. doi: 10.1137/S0036144598347497
  • Bombieri, E. (2000). Problems of the millennium: The Riemann hypothesis. Clay Mathematics Institute. Retrieved from http://www.claymath.org/sites/default/files/official_problem_description.pdf
  • Connes, A. (1999). Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Mathematica (N.S.), 5(1),...
  • Deligne, P. (1974). La conjecture de Weil. I. Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 43(1), 273–307. doi:...
  • Deninger, C. (1998). Some analogies between number theory and dynamical systems on foliated spaces. Documenta Mathematica, Journal der Deutschen...
  • Du Sautoy, M. (2003). The music of the primes. Searching to solve the greatest mystery in mathematics. New York: Harper-Collins Publishers.
  • Euler, L. (1737). Variae observationes circa series infinitas. Commentarii Academiae Scientarium Petropolitanae, 9, 160–188.
  • Katz, N. M., & Sarnak, P. (1999). Random matrices, Frobenius eigenvalues, and monodromy. Providence, Rhode Island: American Mathematical...
  • Lagarias, J. C., & Odlyzko, A. M. (1987). Computing π(x): An analytic method. Journal of Algorithms, 8(2), 173–191. doi: 10.1016/0196-6774(87)90037-x
  • Lapidus, M. L., & Van Frankenhuysen, M. (2001). Dynamical, spectral, and arithmetic zeta functions: AMS special session, San Antonio,...
  • Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function. In Proceedings of Symposia in Pure Mathematcis, XXIV (pp. 181–193)....
  • Odlyzko, A. M. (2001). The 1022-nd zero of the Riemann zeta function. In M. L. Lapidus, & M. van Frankenhuysen (Eds.), Dynamical, spectral,...
  • Oresme, N. (1961). Quaestiones super geometriam Euclidis. Leiden: Brill Archive.
  • Riemann, G. F. B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie, 671–680.
  • Sarnak, P. (2005). Problems of the millennium: The Riemann hypothesis (2004). Clay Mathematics Institute. Retrieved from http://www.claymath.org/library/annual_report/ar2004/04report_prizeproblem.pdf
  • Selberg, A. (1956). Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series....
  • Weil, A. (1949). Numbers of solutions of equations in finite fields. Bulletin of the American Mathematical Society, 55(5), 497–508. doi: 10.1090/S0002-9904-1949-09219-4
  • Weisstein, E. W. (2002). Riemann zeta function zeros. MathWorld–A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html