Abstract
Although the zeta function was first defined and used by Euler, it is to Bernhard Riemann, in an article written in 1859, that we owe our view of the zeta function as a meromorphic function in the plane with a functional equation. Riemann is a very remarkable figure in the history of mathematics. The present article describes his career including the major mathematical highlights, and gives some discussion of his published and unpublished work on the zeta function.
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References
D. Blackbourn, History of Germany 1780–1918, 2nd edn. (Blackwell, Oxford, 2003)
H.M. Bui, B. Conrey, M.P. Young, More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150, 35–64 (2011)
H. Davenport, Multiplicative Number Theory, 3rd edn. (Springer, New York, 2000)
R. Dedekind, The Life of Bernhard Riemann. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 518–533
P.G.L. Dirichlet, Lectures on Number Theory (with supplements by R. Dedekind, translated by J. Stillwell) (American Mathematical Society, Providence, RI/London Mathematical Society, London, 1999)
H.M. Edwards, Riemann’s Zeta Function (Academic, New York, 1974)
C.F. Gauss, Disquisitiones generales circa seriem infinitam, I and II. Werke, Vol. III (Georg Olms Verlag, Hildesheim, 1973/1812)
M. Gromov, Almost flat manifolds. J. Differ. Geom. 13, 223–230 (1978)
A. Ivic, The Riemann Zeta Function. Theory and Applications (Dover Publications, Inc., Mineola, NY, 2003)
E.E. Kummer, Über die Hypergeometrische Reihe. J. Reine Angew. Math. 15, 39–83 and 127–172 (1836); Collected Papers, Vol. II (Springer, Berlin/New York, 1975)
D. Laugwitz, Bernhard Riemann 1826–1866. Turning Points in the Conception of Mathematics (Birkhäuser Boston Inc., Boston, 2008)
A.-M. Legendre, Théorie des Nombres, 2nd edn. (1808). Reproduction, University of Michigan Library, Ann Arbor, MI
S.J. Patterson, An Introduction to the Theory of the Riemann Zeta Function (Cambridge University Press, Cambridge, 1988)
B. Riemann, Foundations for a General Theory of Functions of a Complex Variable. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 1–41
B. Riemann, Contributions to the Theory of the Functions Represented by the Gauss SeriesF(α, β, γ, x). Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 57–75
B. Riemann, Author’s Announcement of the Preceding Paper. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 77–78
B. Riemann, The Theory of Abelian Functions. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 79–134
B. Riemann, On the Number of Primes Less than a Given Magnitude. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 135–143
B. Riemann, On the Propagation of Planar Air Waves of Finite Amplitude. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 145–165
B. Riemann, On the Vanishing ofθ-Functions. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 203–217
B. Riemann, On the Representation of a Function by a Trigonometric Series. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 219–256
B. Riemann, The Hypotheses on Which Geometry is Based. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 257–272
B. Riemann, A Contribution to Electrodynamics. Collected Papers of Bernhard Riemann (translated by R.C. Baker, C. Christensen, H. Orde) (Kendrick Press, Heber City, 2004), pp. 273–278
C.L. Siegel, Über Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik, Abteilung B: Studien, Bd. 2, 45–80
E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edn. (Oxford University Press, Oxford, 1986)
H. Weyl, The Concept of a Riemann Surface, 3rd edn. (translated by G.R. MacLane) (Addison-Wesley, Boston, 1953)
A. Zygmund, Trigonometric Series, vols. I and II (Cambridge University Press, Cambridge, 1959)
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Baker, R. (2017). An Introduction to Riemann’s Life, His Mathematics, and His Work on the Zeta Function. In: Montgomery, H., Nikeghbali, A., Rassias, M. (eds) Exploring the Riemann Zeta Function. Springer, Cham. https://doi.org/10.1007/978-3-319-59969-4_1
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DOI: https://doi.org/10.1007/978-3-319-59969-4_1
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