The Riemann hypothesis: The great pending challenge


Abstract


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


Keywords


prime numbers; zeta function; L function; Riemann hypothesis; millenium problems

References


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. doi: 10.1007/BF01406467

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), 29–106. doi: 10.1007/s000290050042

Deligne, P. (1974). La conjecture de Weil. I. Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 43(1), 273–307. doi: 10.1007/BF02684373

Deninger, C. (1998). Some analogies between number theory and dynamical systems on foliated spaces. Documenta Mathematica, Journal der Deutschen Mathematiker-Vereiningung, Extra Vol. ICM Berlin 1998, 1, 163–186.

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 Society.

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, TX, USA, January 15–16, 1999. Providence, Rhode Island: American Mathematical Society.

Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function. En Proceedings of Symposia in Pure Mathematcis, XXIV (pp. 181–193). Providence, Rhode Island: American Mathematical Society.

Odlyzko, A. M. (2001). The 1022-nd zero of the Riemann zeta function. En M. L. Lapidus, & M. van Frankenhuysen (Eds.), Dynamical, spectral, and arithmetic zeta functions: AMS special session, San Antonio, TX, USA, January 15–16, 1999 (pp. 139–144). Providence, Rhode Island: American Mathematical Society.

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. Journal of the Indian Mathematical Society (N.S.), 20, 47–87.

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







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