Mathematician:Georg Friedrich Bernhard Riemann

Mathematician

German mathematician most famous for the Riemann Hypothesis, which is (at time of writing, early $21$st century) one of the most highly sought-after results in mathematics.

Nationality

German

History

• Born: 17 Sept 1826, Breselenz, Hanover (now Germany)
• 1845: Went to University of Göttingen to study theology, but soon switched to mathematics.
• 1851: Received doctorate from Berlin University
• 1854: Appointed Privatdozent (unpaid lecturer) at Göttingen
• 1855: Replaced Gauss at Göttingen
• 1859: Replaced Dirichlet as full professor
• Died: 20 July 1866, Selasca, Italy

Theorems and Definitions

• Riemann Condition
• (Small) Riemann Function (also known as the Thomae Function or Modified Dirichlet Function)
• Riemannian Geometry
• Riemann Integral
• Riemannian Manifold (also known as a Riemannian Space or Riemann Space)

• Riemannian Metric
  • Pseudo-Riemannian Metric (also known as Semi-Riemannian Metric)

• Riemann P-symbol (also known as the Papperitz symbol for Erwin Papperitz)

• Riemann Sum
  • Upper Riemann Sum (also known as Upper Darboux Sum, for Jean-Gaston Darboux, or Upper Sum)
  • Lower Riemann Sum (also known as Lower Darboux Sum, for Jean-Gaston Darboux, or Lower Sum)

• Riemann Surface
• Riemann Sphere
• Riemann Zeta Function

• Cauchy-Riemann Equations (with Augustin Louis Cauchy)
• Riemann-Christoffel Tensor (with Elwin Bruno Christoffel), also known as Riemannian Curvature Tensor
• Riemann-Stieltjes Integral (with Thomas Joannes Stieltjes)
• Riemann-Stieltjes Sum (with Thomas Joannes Stieltjes)
• Zariski-Riemann Surface (with Oscar Zariski)

• Riemann Hypothesis
• Riemann's Rearrangement Theorem
• Riemann Removable Singularities Theorem
• Riemann Uniformization Theorem

• Grothendieck-Hirzebruch-Riemann-Roch Theorem (with Alexander Grothendieck, Friedrich Hirzebruch and Gustav Roch)
• Hirzebruch-Riemann-Roch Theorem (with Friedrich Hirzebruch and Gustav Roch)

• Riemann-Hilbert Problem (with David Hilbert)
• Riemann-Hurwitz Formula (with Adolf Hurwitz)
• Riemann-Lebesgue Lemma (with Henri Léon Lebesgue)
• Riemann-Lebesgue Theorem (with Henri Léon Lebesgue)
• Riemann-Roch Theorem (with Gustav Roch)
• Riemann-Siegel Formula (with Carl Ludwig Siegel)
• Riemann-Siegel Integral Formula (with Carl Ludwig Siegel)
• Riemann-von Mangoldt Formula (with Hans Carl Friedrich von Mangoldt)

Results named for Georg Friedrich Bernhard Riemann can be found here.

Definitions of concepts named for Georg Friedrich Bernhard Riemann can be found here.

Publications

• 1851: Frundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse
• 1854: Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe
• 1854: Über die Hypothesen, welche der Geometrie zu Grunde liegen
• 1859: Ueber die Anzahl der Primzahlen under einer gegebenen Grösse
• 1925: Die Differential- und Integralgleichungen der Mechanik und Physik (with Philipp Frank, Richard von Mises and Heinrich Weber) (2nd Edition: 1943)

Critical View

... an extraordinary mathematician.

-- Salomon Bochner

The dissertation [ of $1851$ ] submitted by Herr Riemann offers convincing evidence of the author's thorough and penetrating investigations in those parts of the subject treated in the dissertation, of a creative, active, truly mathematical mind, and of a gloriously fertile originality.

-- Carl Friedrich Gauss

Also known as

Usually referred to as Bernhard Riemann.

Sources

• John J. O'Connor and Edmund F. Robertson: "Georg Friedrich Bernhard Riemann": MacTutor History of Mathematics archive

• 1937: Eric Temple Bell: Men of Mathematics: Chapter $\text{XXVI}$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): A List of Mathematicians in Chronological Sequence
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Riemann, Georg Friedrich Bernhard (1826-66)
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): A List of Mathematicians in Chronological Sequence
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): geometry
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Riemann, Georg Friedrich Bernhard (1826-66)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): geometry
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Riemann, Georg Friedrich Bernhard (1826-66)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Riemann, (Georg Friedrich) Bernhard (1826-66)