The Riemann hypothesis: The great pending mathematical challenge

Authors

  • Pilar Bayer University of Barcelona (Spain).

DOI:

https://doi.org/10.7203/metode.0.8903

Keywords:

prime numbers, zeta function, L-function, Riemann hypothesis, millennium problems

Abstract

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.

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Author Biography

Pilar Bayer, University of Barcelona (Spain).

Specialist in number theory. Her academic career is tied to the University of Regensburg (Germany), the Autonomous University of Barcelona, the University of Santander, and the University of Barcelona (the latter three in Spain), where she has resided as a professor since 1982. Her research includes, among other themes, publications on zeta functions, Diophantine equations, elliptic curves, modular forms, and Shimura curves. In the 1980s, she founded the Barcelona Number Theory Seminar, which is still an ongoing program. She has directed fifteen PhD dissertations and is a tenured member of the Royal Spanish Academy of Exact, Physical, and Natural Sciences, the Royal Academy of Sciences and Arts, and the Royal European Academy of Doctors, and is member of the Institute for Catalan Studies. In 2015, she was awarded the Vives University Network Medal of Honour.

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. In Proceedings of Symposia in Pure Mathematics, XXIV (pp. 181–193). Providence, Rhode Island: American Mathematical Society.

Odlyzko, A. M. (2001). The 1022-nd zero of the Riemann zeta function. In 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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Published

2018-06-05

How to Cite

Bayer, P. (2018). The Riemann hypothesis: The great pending mathematical challenge. Metode Science Studies Journal, (8), 35–41. https://doi.org/10.7203/metode.0.8903
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Issue

Making science: A multitude of perspectives (2018)

Section

The millennium problems. Challenges to further mathematics

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